Cauchy Criterion for Real Sequences

Convergence Without a Known Limit

To prove that a sequence converges directly, one often needs to identify its limit and show that its terms approach that particular real number. The Cauchy criterion offers a different route: it tests whether late terms are close to one another, without requiring a candidate limit. That distinction is useful when a sequence is defined by an approximation process and its eventual value is not immediately apparent.

The previous tutorials developed the Cauchy property and proved that every convergent sequence has it. Tutorial 209 stated the converse for real sequences as part of the Cauchy Criterion for Real Sequences. We will use that established criterion rather than repeat its proof. Here we make its quantifiers operational, measure the spread of a sequence’s tail, and examine what a failure of the Cauchy property must look like.

Definition (Cauchy Sequence): A real sequence $(a_n)$ is a Cauchy sequence if, for every $\varepsilon>0$, there exists $N\in\mathbb{N}_0$ such that, for all $m,n\geq N$, $|a_m-a_n|<\varepsilon$.

The order of the quantifiers is important. First, an accuracy $\varepsilon$ is specified. Then one index $N$ is chosen. After that choice, the estimate must hold for every pair of indices $m,n$ in the tail. The indices need not be close to one another: a pair with one index equal to $N$ and the other very large is included.

Theorem (Cauchy Criterion for Real Sequences, recalled): A real sequence converges to a finite real limit if and only if it is a Cauchy sequence.

The criterion depends on working in the real numbers. Its practical message is that, in $\mathbb{R}$, controlling pairwise distances in every sufficiently late tail is enough to guarantee a finite limit, even if that limit has not been found. The forward direction is the result that convergent sequences are Cauchy. The reverse direction is where the completeness of the real numbers matters.

Quantifying the Spread of a Tail

For a bounded sequence, we can summarize all pairwise distances in the tail beginning at $N$ by one number. The sequence is bounded whenever it is Cauchy, by the theorem A Sequence with the Cauchy Property Is Bounded from Tutorial 209. We can therefore use this quantity in particular for every Cauchy sequence.

Definition (Tail Diameter): If $(a_n)$ is a bounded real sequence and $N\in\mathbb{N}_0$, the tail diameter $D_N$ is $$ D_N=\sup\{|a_m-a_n|:m,n\geq N\}. $$

The supremum exists and is finite because the sequence is bounded, so the distances in the set are bounded above. The tail beginning at $N+1$ is contained in the tail beginning at $N$. Consequently, $D_{N+1}\leq D_N$: tail diameters cannot increase as terms are removed.

Theorem (Tail-Diameter Characterization): A bounded real sequence $(a_n)$ is Cauchy if and only if its tail diameters satisfy $D_N\to0$.

Proof. Suppose first that $(a_n)$ is Cauchy. Let $\eta>0$. Apply the Cauchy property with tolerance $\eta/2$. There exists $N$ such that, whenever $m,n\geq N$, $$ |a_m-a_n|<\frac{\eta}{2}. $$ Thus $\eta/2$ is an upper bound for the set defining $D_N$, so $D_N\leq\eta/2<\eta$. For every $K\geq N$, the tail beginning at $K$ is contained in the tail beginning at $N$, and hence $0\leq D_K\leq D_N<\eta$. This proves $D_N\to0$.

Conversely, suppose $D_N\to0$, and let $\varepsilon>0$. There exists $N$ such that $D_N<\varepsilon$. For any $m,n\geq N$, the distance $|a_m-a_n|$ is one of the values whose supremum defines $D_N$. Therefore $$ |a_m-a_n|\leq D_N<\varepsilon. $$ This is exactly the Cauchy property. $\square$

The factor of $1/2$ in the first direction handles a small but important detail: even if every distance in a tail is strictly less than $\eta/2$, the supremum need not itself be strictly less than $\eta/2$. It is enough that the supremum be at most $\eta/2$, which is still strictly less than $\eta$.

Worked Example: Geometric Partial Sums

Let $a_n=\sum_{j=0}^{n}(1/3)^j$. We can test the sequence without first using its eventual limit. If $m>n$, then $$ |a_m-a_n|=\sum_{j=n+1}^{m}\left(\frac13\right)^j =\frac{(1/3)^{n+1}\left(1-(1/3)^{m-n}\right)}{1-1/3} <\frac{(1/3)^{n+1}}{1-1/3} =\frac{1}{2\cdot3^n}. $$

If $n>m$, the same estimate applies after exchanging the indices, and if $m=n$, the distance is zero. Thus, for any $m,n\geq N$, $$ |a_m-a_n|\leq\frac{1}{2\cdot3^N}. $$ Since $1/(2\cdot3^N)\to0$, for every $\varepsilon>0$ we can choose $N$ large enough that $1/(2\cdot3^N)<\varepsilon$. The sequence is Cauchy. By the Cauchy criterion for real sequences, it converges to a finite real limit. This argument establishes convergence without needing to identify the limit first.

What Failure of the Criterion Looks Like

Negating the Cauchy definition gives a useful way to diagnose failure. A sequence is not Cauchy precisely when there is some fixed positive accuracy that it fails to meet in every tail. The separated terms may occur at different indices in different tails, but the size of the required separation does not shrink.

Theorem (Separated Pairs in Every Tail): If a real sequence $(a_n)$ is not Cauchy, then there exists $\varepsilon_0>0$ such that, for every $N\in\mathbb{N}_0$, there are indices $m,n\geq N$ with $|a_m-a_n|\geq\varepsilon_0$. Moreover, there are strictly increasing index sequences $(p_k)$ and $(q_k)$ such that $|a_{p_k}-a_{q_k}|\geq\varepsilon_0$ for every $k$.

Proof. Since $(a_n)$ is not Cauchy, the negation of the definition supplies an $\varepsilon_0>0$ such that for every $N$, there are $m,n\geq N$ with $|a_m-a_n|\geq\varepsilon_0$. We now choose pairs recursively. Start with any $N_0$, and choose a pair of indices at least $N_0$ satisfying the separation. Write the smaller index as $p_0$ and the larger as $q_0$. Having chosen $p_k,q_k$, set $N_{k+1}=\max\{p_k,q_k\}+1$. Apply the separation property at $N_{k+1}$, and again write the resulting smaller and larger indices as $p_{k+1}$ and $q_{k+1}$.

Both new indices exceed $\max\{p_k,q_k\}$. In particular, $p_{k+1}>p_k$ and $q_{k+1}>q_k$, so both index sequences are strictly increasing. At every stage the selected pair satisfies $|a_{p_k}-a_{q_k}|\geq\varepsilon_0$. This proves the claim. $\square$

The fixed separation also explains why a non-Cauchy sequence cannot converge. If the sequence converged, the result that convergent sequences are Cauchy would apply. The separated-pairs description gives more information than simply knowing convergence fails: no matter how far out one goes, the tail contains two terms at least $\varepsilon_0$ apart.

Worked Example: An Alternating Sequence Is Not Cauchy

Consider $a_n=(-1)^n$. Given any $N$, choose an even index $m\geq N$ and an odd index $n\geq N$. Then $a_m=1$ and $a_n=-1$, so $$ |a_m-a_n|=|1-(-1)|=2. $$ Thus the same separation $\varepsilon_0=1$, for example, works in every tail, because $2\geq1$. The separated-pairs theorem confirms that the sequence is not Cauchy. By the Cauchy criterion, it does not converge in $\mathbb{R}$.

Applying Pairwise Estimates

In many examples, the most effective approach is to find a bound on $|a_m-a_n|$ that becomes small when both indices are large. The bound need not be the sharpest possible. It must, however, apply to every pair in the tail and tend to zero as the tail begins later.

Worked Example: An Alternating Error Around a Constant

Define $b_n=5+(-1)^n/(n+2)$. For any $m,n\in\mathbb{N}_0$, the triangle inequality gives $$ |b_m-b_n| =\left|\frac{(-1)^m}{m+2}-\frac{(-1)^n}{n+2}\right| \leq\frac{1}{m+2}+\frac{1}{n+2}. $$ If $m,n\geq N$, then $m+2\geq N+2$ and $n+2\geq N+2$. Therefore $$ |b_m-b_n|\leq\frac{2}{N+2}. $$ Given $\varepsilon>0$, choose $N=\lfloor2/\varepsilon\rfloor+1$. Then $N+2>2/\varepsilon$, so $2/(N+2)<\varepsilon$. Hence $(b_n)$ is Cauchy and therefore converges in $\mathbb{R}$. The alternating signs cause no difficulty: the estimate uses absolute values and holds for every pair of indices.

Worked Example: A Growing Sequence Fails the Cauchy Test

Let $c_n=n$. For any $N$, choose $m=N$ and $n=N+2$. Both indices are at least $N$, but $$ |c_m-c_n|=|N-(N+2)|=2. $$ Thus, with $\varepsilon_0=1$, every tail contains two terms whose distance is at least $\varepsilon_0$. The sequence is not Cauchy and hence cannot converge to a finite real number. Notice that it is not necessary to compare the sequence with a proposed limit; the pairwise test alone rules out convergence.

Choosing the Right Test

The Cauchy criterion is most useful when a direct estimate between two terms is simpler than an estimate between a term and a proposed limit. Typical strategies include expressing $a_m-a_n$ as a finite sum and bounding its terms, or comparing each of $a_m$ and $a_n$ to a common reference value that may depend on the indices. In either case, the final bound must become arbitrarily small when both indices are restricted to a sufficiently late tail.

A common mistake is to show that consecutive terms are close and infer that the sequence is Cauchy. Small successive differences alone do not guarantee that distant terms in the same tail are close: for instance, the sequence $c_n=n$ has successive differences equal to $1$, yet terms far apart in a tail can be arbitrarily far apart. A valid Cauchy argument controls arbitrary $m,n\geq N$, not only $a_{n+1}$ and $a_n$.

Another useful distinction is between proving the Cauchy property and proving convergence by the criterion. The Cauchy estimate is a statement about all late pairs. Once it is established, the Cauchy criterion guarantees a real limit, but it does not by itself name that limit. If the problem also asks for the value, one may need a separate argument to identify it.

Check Your Understanding

Use the quantifiers, tail-diameter characterization, and separated-pairs result to answer the following questions.

In the definition of a Cauchy sequence, why must the chosen index work for every pair $m,n\geq N$?
Why does the proof that a Cauchy sequence has tail diameters tending to zero use $\eta/2$ rather than $\eta$?
What fixed separation must exist in every tail if a sequence is not Cauchy?
For the geometric partial sums in the example, what feature of the estimate on $|a_m-a_n|$ makes it useful?
Why is control of $|a_{n+1}-a_n|$ alone not enough to establish the Cauchy property?