# Convergent Sequence is Cauchy Sequence

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## Contents

## Theorem

Let $M = \left({A, d}\right)$ be a metric space.

Every convergent sequence in $M$ is a Cauchy sequence.

## Proof

Let $\left \langle {x_n} \right \rangle$ be a sequence in $A$ that converges to the limit $l \in A$.

Let $\epsilon > 0$.

Then also $\dfrac \epsilon 2 > 0$.

Because $\left \langle {x_n} \right \rangle$ converges to $l$, we have:

- $\exists N: \forall n > N: d \left({x_n, l}\right) < \dfrac \epsilon 2$

In the same way:

- $\forall m > N: d \left({x_m, l}\right) < \dfrac \epsilon 2$

So if $m > N$ and $n > N$, then:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle d \left({x_n, x_m}\right)\) | \(\le\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle d \left({x_n, l}\right) + d \left({l, x_m}\right)\) | \(\displaystyle \) | \(\displaystyle \) | (by the Triangle Inequality) | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(<\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \frac \epsilon 2 + \frac \epsilon 2\) | \(\displaystyle \) | \(\displaystyle \) | (by choice of $N$) | ||

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \) | \(\) | \(\displaystyle \) | \(\displaystyle \epsilon\) | \(\displaystyle \) | \(\displaystyle \) |

Thus $\left \langle {x_n} \right \rangle$ is a Cauchy sequence.

$\blacksquare$

## Also see

- Complete Metric Space, where the converse is true.

## Sources

- Walter Rudin:
*Principles of Mathematical Analysis*(1953)... (next): $3.11a$ - Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(1970)... (previous)... (next): $\text{I}: \ \S 5$: Complete Metric Spaces - W.A. Sutherland:
*Introduction to Metric and Topological Spaces*(1975)... (previous)... (next): $7.2$: Sequential compactness: Proposition $7.2.4$ - K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*(1977)... (previous)... (next): $\S 5.17$ - K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*(1977)... (previous)... (next): Appendix: $\S 18.4$: Subsequences