Convergent Sequence is Cauchy Sequence/Metric Space
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Every convergent sequence in $M$ is a Cauchy sequence.
Proof
Let $\sequence {x_n}$ be a sequence in $A$ that converges to the limit $l \in A$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ converges to $l$, we have:
- $\exists N: \forall n > N: \map d {x_n, l} < \dfrac \epsilon 2$
So if $m > N$ and $n > N$, then:
\(\ds \map d {x_n, x_m}\) | \(\le\) | \(\ds \map d {x_n, l} + \map d {l, x_m}\) | Triangle Inequality | |||||||||||
\(\ds \) | \(<\) | \(\ds \frac \epsilon 2 + \frac \epsilon 2\) | (by choice of $N$) | |||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
Thus $\sequence {x_n}$ is a Cauchy sequence.
$\blacksquare$
Also see
- Definition:Complete Metric Space, where the converse is true.
Sources
- 1953: Walter Rudin: Principles of Mathematical Analysis ... (next): $3.11a$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $7.2$: Sequential compactness: Proposition $7.2.4$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $4$: Complete Normed Spaces