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Theorem List for Metamath Proof Explorer - 12101-12200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlo1mul 12101* The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1mul2 12102* The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1dif 12103* If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1sub 12104* The difference of an eventually upper bounded function and an eventually bounded function is eventually upper bounded. The "correct" sharp result here takes the second function to be eventually lower bounded instead of just bounded, but our notation for this is simply , so it is just a special case of lo1add 12100. (Contributed by Mario Carneiro, 31-May-2016.)

Theoremclimadd 12105* Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)

Theoremclimmul 12106* Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)

Theoremclimsub 12107* Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)

Theoremclimaddc1 12108* Limit of a constant added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremclimaddc2 12109* Limit of a constant added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremclimmulc2 12110* Limit of a sequence multiplied by a constant . Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremclimsubc1 12111* Limit of a constant subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremclimsubc2 12112* Limit of a constant minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Theoremclimle 12113* Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)

Theoremclimsqz 12114* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremclimsqz2 12115* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremrlimadd 12116* Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremrlimsub 12117* Limit of the difference of two converging functions. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremrlimmul 12118* Limit of the product of two converging functions. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremrlimdiv 12119* Limit of the quotient of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremrlimneg 12120* Limit of the negative of a sequence. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremrlimle 12121* Comparison of the limits of two sequences. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimsqzlem 12122* Lemma for rlimsqz 12123 and rlimsqz2 12124. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)

Theoremrlimsqz 12123* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)

Theoremrlimsqz2 12124* Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 3-Feb-2014.) (Revised by Mario Carneiro, 20-May-2016.)

Theoremlo1le 12125* Transfer eventual upper boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1le 12126* Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremrlimno1 12127* A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremclim2ser 12128* The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)

Theoremclim2ser2 12129* The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)

Theoremiserex 12130* An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)

Theoremisermulc2 12131* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)

Theoremclimlec2 12132* Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)

Theoremiserle 12133* Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremiserge0 12134* The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremclimub 12135* The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)

Theoremclimserle 12136* The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Theoremisershft 12137 Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremisercolllem1 12138* Lemma for isercoll 12141. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremisercolllem2 12139* Lemma for isercoll 12141. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremisercolllem3 12140* Lemma for isercoll 12141. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremisercoll 12141* Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremisercoll2 12142* Generalize isercoll 12141 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremclimsup 12143* A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)

Theoremclimcau 12144* A converging sequence of complex numbers is a Cauchy sequence. Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)

Theoremcaucvgrlem 12145* Lemma for caurcvgr 12146. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremcaurcvgr 12146* A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.)

Theoremcaucvgrlem2 12147* Lemma for caucvgr 12148. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.)

Theoremcaucvgr 12148* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremcaurcvg 12149* A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremcaurcvg2 12150* A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.)

Theoremcaucvg 12151* A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)

Theoremcaucvgb 12152* A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.)

Theoremserf0 12153* If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)

Theoremiseraltlem1 12154* Lemma for iseralt 12157. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremiseraltlem2 12155* Lemma for iseralt 12157. The terms of an alternating series form a chain of inequalities in alternate terms, so that for example and (assuming so that these terms are defined). (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremiseraltlem3 12156* Lemma for iseralt 12157. From iseraltlem2 12155, we have and , and we also have for each by the definition of the partial sum , so combining the inequalities we get , so and . Thus both even and odd partial sums are Cauchy if converges to . (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremiseralt 12157* The alternating series test. If is a decreasing sequence that converges to , then is a convergent series. (Note that the first term is positive if is even, and negative if is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by using isermulc2 12131.) (Contributed by Mario Carneiro, 7-Apr-2015.)

5.8.3  Finite and infinite sums

Syntaxcsu 12158 Extend class notation to include finite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is is a commonly used word in comments.)

Definitiondf-sum 12159* Define the sum of a series with an index set of integers . is normally a free variable in , i.e. can be thought of as . This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e. finite subsets of the upper integers) by summo 12190. Examples: means , and means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12338). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumex 12160 A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumeq1f 12161 Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumeq1 12162* Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremnfsum1 12163* Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremnfsum 12164* Bound-variable hypothesis builder for sum: if is (effectively) not free in and , it is not free in . (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumeq2w 12165* Equality theorem for sum, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremsumeq2ii 12166* Equality theorem for sum, with the class expressions and guarded by to be always sets. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremsumeq2 12167* Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremcbvsum 12168* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremcbvsumv 12169* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremcbvsumi 12170* Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.)

Theoremsumeq1i 12171* Equality inference for sum. (Contributed by NM, 2-Jan-2006.)

Theoremsumeq2i 12172* Equality inference for sum. (Contributed by NM, 3-Dec-2005.)

Theoremsumeq12i 12173* Equality inference for sum. (Contributed by FL, 10-Dec-2006.)

Theoremsumeq1d 12174* Equality deduction for sum. (Contributed by NM, 1-Nov-2005.)

Theoremsumeq2d 12175* Equality deduction for sum. Note that unlike sumeq2dv 12176, may occur in . (Contributed by NM, 1-Nov-2005.)

Theoremsumeq2dv 12176* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremsumeq2sdv 12177* Equality deduction for sum. (Contributed by NM, 3-Jan-2006.)

Theorem2sumeq2dv 12178* Equality deduction for double sum. (Contributed by NM, 3-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremsumeq12dv 12179* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)

Theoremsumeq12rdv 12180* Equality deduction for sum. (Contributed by NM, 1-Dec-2005.)

Theoremsum2id 12181* The second class argument to a sum can be chosen so that it is always a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Theoremsumfc 12182* A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Theoremfz1f1o 12183* A lemma for working with finite sums. (Contributed by Mario Carneiro, 22-Apr-2014.)

Theoremsumrblem 12184* Lemma for sumrb 12186. (Contributed by Mario Carneiro, 12-Aug-2013.)

Theoremfsumcvg 12185* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)

Theoremsumrb 12186* Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 9-Apr-2014.)

Theoremsummolem3 12187* Lemma for summo 12190. (Contributed by Mario Carneiro, 29-Mar-2014.)

Theoremsummolem2a 12188* Lemma for summo 12190. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 20-Apr-2014.)

Theoremsummolem2 12189* Lemma for summo 12190. (Contributed by Mario Carneiro, 3-Apr-2014.)

Theoremsummo 12190* A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremzsum 12191* Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 9-Apr-2014.)

Theoremisum 12192* Series sum with an upper integer index set (i.e. an infinite series). (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 7-Apr-2014.)

Theoremfsum 12193* The value of a sum over a nonempty finite set. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremsum0 12194 Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)

Theoremsumz 12195* Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)

Theoremfsumf1o 12196* Re-index a finite sum using a bijection. (Contributed by Mario Carneiro, 20-Apr-2014.)

Theoremsumss 12197* Change the index set to a subset in an upper integer sum. (Contributed by Mario Carneiro, 21-Apr-2014.)

Theoremfsumss 12198* Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)

Theoremsumss2 12199* Change the index set of a sum by adding zeroes. (Contributed by Mario Carneiro, 15-Jul-2013.) (Revised by Mario Carneiro, 20-Apr-2014.)

Theoremfsumcvg2 12200* The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.)

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