Cauchy's Criterion

Cauchy's criterion is a well known criteria for when an infinite sum converges, that is, has a finite limit.

Augustin Cauchy was not the first to come up with the criteria. Leonhard Euler, for example, used a similiar criteria. It may be that the significance of the criteria became appreciated in the context of Cauchy's great work on the foundations of calculus.

Definition 1: Cauchy Sequence

A sequence s_i is Cauchy Sequence if and only if given any positive number ε, there exists an integer N such that if m,n are greater than N, then absolute(s_m - s_n) is less than ε

In other words, elements of the sequence get arbitrarily close to one another.

I will need a few properties of absolute inequalities:

Lemma 1: absolute(a - b) ≤ absolute(a) + absolute(b)

Proof:

(1) Case I: a - b is nonnegative

So abs(a-b) = a - b

If b ≥ 0, then a - b ≤ a ≤ abs(a) + abs(b)

If b is less than 0, then a - b = a + abs(b) ≤ abs(a) + abs(b)

(2) Case II: a - b is negative

So abs(a-b) = -(a-b) = b - a = abs(b - a)

Using step #1, we know that abs(b - a) ≤ abs(b) + abs(a) = abs(a) + abs(b)

So that:

abs(a - b) ≤ abs(a) + abs(b)

QED

Lemma 2: abs(a) - abs(b) ≤ abs(a - b)

Proof:

(1) Case I: a - b is nonnegative so that abs(a - b) = a - b

If a ≥ 0, then abs(a - b) = a - b ≥ abs(a) - abs(b)

NOTE: It is = except for the case where b is negative.

If a is less than 0, then abs(b) is greater than abs(a) and abs(a) - abs(b) must be a negative number.

(2) Case II: a - b is negative and abs(a - b) = -(a - b) = b - a

If a is ≥ 0, then abs(b) is greater than a and abs(a) - abs(b) is a negative number.

If a is less than 0 and b is less than 0, then b - a = b + abs(a) = abs(a) - abs(b) so that abs(a - b) = abs(a) - abs(b)

If a is less than 0 and b ≥ 0, then b - a = b + abs(a) = abs(a) + abs(b) ≥ abs(a) - abs(b).

QED

Here are some properties of Cauchy Sequences which I will use below:

Lemma 3: Any convergent sequences is a Cauchy Sequence

Proof:

(1) Let a_i be a convergent sequence (that is, as a_i gets larger, it approaches a limit) so that it's limit = L. [See definition 7, here for definition of a convergent sequence]

(2) So from the above definition, we know for any positive number ε, there exists a positive number N such that:

if n is greater than N, then absolute(a_n - L) is less than ε

(3) So, for a value (1/2)ε, there exists an integer N such that if n is greater than N, absolute(a_n - L) is less than (1/2)ε

(3) So let's assume that we have two integers m,n both greater than N.

(4) This means that in both cases absolute(a_m - L) is less than (1/2)ε and absolute(a_n - L) is less than (1/2)ε

(5) This gives us:

absolute(a_m - a_n) = absolute([a_m - L] - [a_n - L]) ≤ absolute(a_m - L) + absolute(a_n - L) [See Lemma 1 above]

(6) Finally,

absolute(a_m - L) - absolute(a_n - L) is less than (1/2)ε + (1/2)ε = ε

QED

Lemma 4: A Cauchy Sequence has a bound

Proof:

(1) Let (a_i) be a Cauchy Sequence.

(2) Then, for any positive number ε greater than 0, there is an integer N such that:

for any integer m,n ≥ N, abs(a_n - a_m) is less than ε [See Definition of a Cauchy Sequence above]

(3) So that if we ε = 1 (since ε can be any positive number), we have:

abs(a_n) - abs(a_m) ≤ abs(a_n - a_m) less than 1 for all n,m ≥ N. [See Lemma 2 above]

(4) Let m = N (since m can be any integer ≥ N), then we have:

abs(a_n) - abs(a_N) is less than 1 which means that:

abs(a_n) is less than abs(a_N) + 1 for n ≥ N.

(5) Now, let n = N (since n can be any integer ≥ N), then we have:

abs(a_N) - abs(a_m) is less than 1 which means that:

abs(a_m) is greater than abs(a_N) - 1 for all n ≥ N.

(6) So, for all n, we have:

abs(a_n) is less than max { abs(a₁), ..., abs(a_N-1), abs(a_N) + 1 }

and

abs(a_n) is greater than min { abs(a_n1), ..., abs(a_N-1), abs(a_N) - 1 }

(7) This shows that for all finite subsets of the sequence, there exists a bound for a_i where upper bound = max { abs(a₁), ..., abs(a_N-1), abs(a_N) + 1 } and a lower bound = min { abs(a₁, ..., abs(a_N-1), abs(a_N)-1 }

QED

Lemma 5: If a Cauchy Sequence has a subsequence convergent to b, then the Cauchy sequence itself converges to b.

Proof:

(1) Let a_n be a Cauchy sequence with the subsequence a_in convergent to b.

(2) By the definition of convergence, we know that for a positive number ε/2, there exists an integer M such that for all n ≥ M abs(a_in - b) ≤ ε/2.

(3) By the definition of a Cauchy sequence, we know that there exists an integer n₀ such that for all m,n ≥ n₀, abs(a_n - a_m) ≤ ε/2.

(4) Now, if i_M (this is the start of the subsequence that converges) is less than n₀, we can always find a M' which is greater than M such that i_M' ≥ n₀.

We can assume this since we are assuming an infinite subsequence.

(5) So for all n ≥ n₀, we have:

abs(a_n - b) = abs(a_n - a_iM' + a_iM' - b) ≤ abs(a_n - a_iM') + abs(a_iM' - b) [See Lemma 1 above]

(6) Now, abs(a_n - a_iM') + abs(a_iM' - b) is less than ε/2 + ε/2 = ε

We know that abs(a_n - a_iM') is less than ε/2 from the definition of a Cauchy sequence.

We know that abs(a_iM') is less than ε/2 from the definition of the convergent sequence.

(7) Now putting this all together gives us that:

abs(a_n - b) ≤ ε

Which by definition (see Definition 7, here) means that lim(a_n) = b.

QED

Lemma 6: Every real Cauchy sequence is convergent.

Proof:

(1) By Lemma 4 above, every Cauchy sequence is bounded.

(2) So, by the Bolzano-Weierstrass Theorem (see Theorem, here), every Cauchy sequence has a convergent subsequence.

(3) So, by Lemma 5 above, every Cauchy sequence is convergent.

QED

Lemma 7: A sequence of reals converges if and only if it is a Cauchy sequence

Proof:

(1) By Lemma 6 above, we know that a Cauchy sequence is convergent.

(2) By Lemma 3 above, we know that a convergent sequence is a Cauchy sequence.

QED

Here is the Criterion:

Theorem: Cauchy's Criterion

A series a_i is convergent (that is, has a finite limit) if and only if for every positive number ε, there exists a positive integer N such that:

for all n greater than N and p ≥ 1:

absolute(a_n+1 + a_n+2 + ... + a_n+p) is less than ε

Proof:

(1) Let s_n = ∑ a_i

The assumption here is that i ranges from 0 to n.

(2) s_n converges if and only if it is a Cauchy Sequence. [See Lemma 7 above]

(3) Assume than s_n is a Cauchy Sequence.

(4) Then, for every positive number ε, there exists a number N such that for all integers n,m greater than N, absolute(s_m - s_n) is less than ε [See Definition 1 above]

(5) Let's assume that m is greater than n.

At this point, we've made no assumption about m or n and this is consistent with our assumption in step #3.

(6) We know that there exists an integer p ≥ 1 such that m = n + p

(7) Based on the definition of s_n, we can see that:

absolute(s_m - s_n) = absolute(s_n+p - s_n) = absolute(a_n+1 + a_n+2 + ... + a_n+p)

(8) Now, using step #6, we can see that ∑ a_i is convergent if and only if the conditions of the given apply.

QED

References

• Somsack Chaitesipaseut, Cauchy's Criterion for Convergence
• Cauchy's Criterion for Convergence, PlanetMath.org
• Cauchy Sequences
• Cauchy Sequences and the Completeness of the Reals, Mathology