Proof of the Cauchy criterion for series

Builds on Cauchy criterion for sequences Cauchy sequence Partial sum Convergent series

Proof

We apply the Cauchy criterion to the sequence of partial sums.

Let $s_n = \sum_{k=0}^{n} a_k$ be the partial sums. For natural numbers $n > m$ , we have $s_n - s_m = \sum_{k=m+1}^{n} a_k$ .
By definition, $\sum a_n$ is a convergent series exactly when $(s_n)$ converges.
By the Cauchy criterion for sequences, $(s_n)$ converges if and only if it is a Cauchy sequence.
Assume that $(s_n)$ is Cauchy, and let $\varepsilon > 0$ . There is $N$ with $\abs{s_n - s_m} < \varepsilon$ for all $m, n \ge N$ . For $n > m \ge N$ , step 1 turns this into $\abs{\sum_{k=m+1}^{n} a_k} < \varepsilon$ .
Conversely, assume the condition of the statement, and let $\varepsilon > 0$ . Take $N$ as in the condition, and let $m, n \ge N$ .
If $m = n$ , then $\abs{s_n - s_m} = 0 < \varepsilon$ . If $n > m$ , step 1 and the condition give $\abs{s_n - s_m} < \varepsilon$ . If $m > n$ , swap the roles of $m$ and $n$ .
So $(s_n)$ is a Cauchy sequence.
By steps 2 to 7, the series converges if and only if the condition holds.

Idea

A block sum $a_{m+1} + \cdots + a_n$ is the difference $s_n - s_m$ of two partial sums. So the condition says exactly that the partial sums form a Cauchy sequence, and Cauchy sequences of reals converge.

Reading path

To understand this page from the ground up, read these 80 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof that convergent sequences are Cauchy, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of strong induction from ordinary induction, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof that a decreasing sequence bounded below converges, Proof of the monotone convergence theorem for sequences.

Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Integers object
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Set definition
Ring of integers statement
Successor function definition
Implication definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Ordered pair definition
Addition of natural numbers definition
Universal quantifier definition
The successor function is injective statement
Zero is not a successor statement
The rationals form a field statement
Bounded sequence definition
Cauchy sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Completeness axiom statement
Cartesian product definition
Subset definition
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
The rationals form an ordered field statement
Convergent sequence definition
Partial sum definition
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Convergent series definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Strong induction statement
Preorder definition
Proof of the well-ordering principle from strong induction proof
Partial order definition
Well-ordering principle statement
Total order definition
Proof that every sequence has a monotone subsequence, by peaks proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Epsilon-N definition of the limit of a sequence statement
Proof that convergent sequences are Cauchy proof
Proof that Cauchy sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof that a decreasing sequence bounded below converges proof
Decreasing sequence bounded below converges statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bolzano–Weierstrass theorem statement
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Cauchy criterion for sequences statement
Proof of the Cauchy criterion for series proof

Where this leads

507 pages build on this one.

Tag	`80XPRSD36C`
URI	`m2://80XPRSD36C`
Kind	proof
Writing	W0 stub
MSC	40A05
Proves	Cauchy criterion for series
Uses	Cauchy criterion for sequences, Cauchy sequence, Partial sum, Convergent series
Changed	4 November 2026

Proof of the Cauchy criterion for series

Proof

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Reading path

Where this leads

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