Convergent Sequences Are Cauchy

From a Limit to Tail Closeness

The previous tutorial developed the Cauchy property as a condition on pairs of terms: after some index, every pair in the tail must be close. Earlier in this course, Tutorial 209 established the Cauchy Criterion for Real Sequences, including the result that every convergent real sequence is Cauchy. We will use that theorem rather than prove it again. Here the emphasis is quantitative: when we know how quickly a sequence approaches its limit, what can we say about the distance between two late terms?

The key estimate compares each term with the same limit. If \(a_n\) is close to \(L\) and \(a_m\) is close to \(L\), then the triangle inequality gives \(|a_m-a_n|\leq |a_m-L|+|a_n-L|\). A rate of convergence therefore supplies an explicit index after which all pairwise distances are small. Such an index is useful when a proof needs more than the general fact that a sequence is Cauchy.

Definition (Convergence Modulus): Suppose \(a_n\to L\). A function \(M:(0,\infty)\to\mathbb{N}_0\) is a convergence modulus for \((a_n)\) and \(L\) if, for every \(\delta>0\) and every \(n\geq M(\delta)\), \(|a_n-L|<\delta\).

A convergence modulus records a valid choice of the index in the epsilon–N definition, as a function of the requested accuracy. It need not be the smallest possible choice. When an explicit error estimate is available, it often gives a convenient modulus directly.

Theorem (A Convergence Modulus Gives a Cauchy Modulus): Suppose \(a_n\to L\), and let \(M\) be a convergence modulus. For every \(\varepsilon>0\), all \(m,n\geq M(\varepsilon/2)\) satisfy \(|a_m-a_n|<\varepsilon\). Thus \(M(\varepsilon/2)\) is an explicit Cauchy index for tolerance \(\varepsilon\).

Proof. Fix \(\varepsilon>0\), and set \(N=M(\varepsilon/2)\). If \(m,n\geq N\), the definition of convergence modulus gives both \(|a_m-L|<\varepsilon/2\) and \(|a_n-L|<\varepsilon/2\). By the triangle inequality,

|a_m-a_n| =|(a_m-L)+(L-a_n)| \leq |a_m-L|+|a_n-L| <\frac{\varepsilon}{2}+\frac{\varepsilon}{2} =\varepsilon.

The same \(N\) works for every pair \(m,n\geq N\), which is precisely the required tail estimate. The Cauchy property itself was established earlier; this result explains how an available convergence modulus produces a quantitative index for it. \(\square\)

Worked Estimates from Error Bounds

In practice, one may know a direct estimate on the error rather than a modulus written as a function. If \(|a_n-L|\leq r_n\), then for any two indices \(m,n\), \(|a_m-a_n|\leq r_m+r_n\). If the bounds \(r_n\) decrease, a single tail bound can control both terms at once.

Corollary (Pairwise Bound from a Decreasing Error Bound): Suppose \(|a_n-L|\leq r_n\) for every \(n\), where \(r_n\geq0\) and \((r_n)\) is nonincreasing. Then for every \(m,n\geq N\), \(|a_m-a_n|\leq 2r_N\).

Proof. If \(m,n\geq N\), nonincreasingness gives \(r_m\leq r_N\) and \(r_n\leq r_N\). Applying the triangle inequality through \(L\),

|a_m-a_n| \leq |a_m-L|+|a_n-L| \leq r_m+r_n \leq 2r_N.

This bound is valid for every pair in the tail, including pairs whose indices are far apart. \(\square\)

Worked Example: An Explicit Index for a Reciprocal Error

Let \(a_n=5+\frac{2}{n+1}\), with \(n\in\mathbb{N}_0\). Its limit is \(5\), and the error is exactly \(|a_n-5|=2/(n+1)\). For any \(\delta>0\), choose \(M(\delta)=\lfloor 2/\delta\rfloor+1\), where \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\). Then \(M(\delta)+1>2/\delta\), so whenever \(n\geq M(\delta)\),

|a_n-5|=\frac{2}{n+1} \leq \frac{2}{M(\delta)+1} <\delta.

Thus this \(M\) is a convergence modulus. For a pairwise tolerance \(\varepsilon>0\), the theorem gives the Cauchy index \(N=M(\varepsilon/2)=\lfloor 4/\varepsilon\rfloor+1\). Directly, if \(m,n\geq N\), then

|a_m-a_n| =2\left|\frac{1}{m+1}-\frac{1}{n+1}\right| \leq \frac{2}{m+1}+\frac{2}{n+1} \leq \frac{4}{N+1} <\varepsilon.

This calculation gives a usable index rather than merely asserting that some index exists. The estimate is deliberately uniform in \(m\) and \(n\); it does not depend on the gap between their indices.

Worked Example: Alternating Terms with a Reciprocal Error

Consider \(b_n=\frac{(-1)^n}{n+2}\). Since \(|b_n-0|=1/(n+2)\), for \(\delta>0\) one may take \(M(\delta)=\lfloor 1/\delta\rfloor\). Indeed, \(M(\delta)+2>1/\delta\), and for every \(n\geq M(\delta)\),

|b_n|=\frac{1}{n+2} \leq\frac{1}{M(\delta)+2} <\delta.

The Cauchy index supplied by the modulus for tolerance \(\varepsilon\) is \(N=\lfloor 2/\varepsilon\rfloor\). For \(m,n\geq N\), including when the signs of \(b_m\) and \(b_n\) differ, the estimate is

|b_m-b_n| \leq |b_m|+|b_n| =\frac{1}{m+2}+\frac{1}{n+2} \leq\frac{2}{N+2} <\varepsilon.

The sign alternation does not obstruct the estimate: comparison with the common limit \(0\) handles both signs at once. The final strict inequality follows from \(N+2>2/\varepsilon\).

Comparing Tails of Two Convergent Sequences

The same approach can compare terms from two different sequences, even when their indices are chosen independently. This is useful when two approximations are being compared, or when one wants to show that late terms from different sequences lie near the same target.

Theorem (Cross-Tail Estimate for Two Limits): Suppose \(a_n\to A\) and \(b_n\to B\), with convergence moduli \(M_a\) and \(M_b\). For any \(\alpha,\beta>0\), if \(m\geq M_a(\alpha)\) and \(n\geq M_b(\beta)\), then \(|a_m-b_n|<\alpha+|A-B|+\beta\).

Proof. The modulus conditions give \(|a_m-A|<\alpha\) and \(|b_n-B|<\beta\). Insert the two limits between the terms and apply the triangle inequality:

|a_m-b_n| =|(a_m-A)+(A-B)+(B-b_n)| \leq |a_m-A|+|A-B|+|B-b_n| <\alpha+|A-B|+\beta.

The indices \(m\) and \(n\) need not be equal. The estimate controls every eligible pair of terms, and the middle quantity \(|A-B|\) records the separation between the limits. \(\square\)

Worked Example: Tails Approaching Different Limits

Let \(x_n=2+\frac{1}{n+1}\) and \(y_n=-1+\frac{2}{n+2}\). Their limits are \(2\) and \(-1\), respectively. For \(\alpha,\beta>0\), valid convergence moduli are \(M_x(\alpha)=\lfloor1/\alpha\rfloor\) and \(M_y(\beta)=\lfloor2/\beta\rfloor\). To check the first, \(M_x(\alpha)+1>1/\alpha\), so \(1/(n+1)<\alpha\) for \(n\geq M_x(\alpha)\). For the second, \(M_y(\beta)+2>2/\beta\), so \(2/(n+2)<\beta\) for \(n\geq M_y(\beta)\).

Since the limits are three units apart, the cross-tail estimate gives, whenever \(m\geq M_x(\alpha)\) and \(n\geq M_y(\beta)\),

|x_m-y_n| <\alpha+|2-(-1)|+\beta =\alpha+3+\beta.

For a concrete choice, take \(\alpha=\beta=1/4\). Then \(M_x(1/4)=4\) and \(M_y(1/4)=8\), so every \(m\geq4\) and \(n\geq8\) satisfy \(|x_m-y_n|<7/2\). The estimate also reflects why the two tails do not become arbitrarily close to one another: their limits remain three units apart.

What the Estimates Do—and Do Not—Say

The general theorem that convergent real sequences are Cauchy guarantees tail closeness without requiring an explicit rate. A convergence modulus strengthens that conclusion by specifying an index for each tolerance. When no convenient formula for a modulus is available, the qualitative theorem still applies; when a formula is available, it can make later estimates substantially more precise.

A common mistake is to confuse small successive differences with the Cauchy property. The Cauchy condition requires one index that controls every pair in the tail, not just neighboring terms. For example, \(c_n=\sqrt{n}\) has successive differences \[ c_{n+1}-c_n=\frac{1}{\sqrt{n+1}+\sqrt n}, \] which tend to zero, but the sequence is not Cauchy. For any proposed \(N\), choose \(r=\max\{N,1\}\), \(n=r\), and \(m=4r\). Then \[ |c_m-c_n|=\sqrt{4r}-\sqrt r=\sqrt r\geq1. \] So no tail has all its terms within distance \(1\) of one another. Convergence avoids this problem because closeness to a single limit controls all pairs at once.

The cross-tail estimate also makes the role of the limits explicit. If \(A=B\), choosing small \(\alpha\) and \(\beta\) makes terms from the two tails close, even when their indices differ. If \(A\neq B\), the term \(|A-B|\) remains in the bound: the tails approach different locations, so the estimate should not suggest that the sequences become arbitrarily close to each other.

Check Your Understanding

Use the definitions and estimates in this tutorial to answer the following questions.

What information does a convergence modulus provide that the statement \(a_n\to L\) alone does not specify numerically?
Why does the theorem use the convergence tolerance \(\varepsilon/2\) to obtain a pairwise tolerance \(\varepsilon\)?
For a nonincreasing error bound \(r_n\), why does \(r_m+r_n\leq2r_N\) hold when \(m,n\geq N\)?
In the cross-tail estimate, what does the term \(|A-B|\) represent?
Why are small successive differences insufficient to prove that a sequence is Cauchy?