Real Convergent Sequence is Cauchy Sequence

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Theorem

Every convergent real sequence in $\R$ is a Cauchy sequence.


Proof 1

Let $\sequence {x_n}$ be a sequence in $\R$ that converges to the limit $l \in \R$.

Let $\epsilon > 0$.

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

Because $\sequence {x_n}$ converges to $l$, we have:

$\exists N: \forall n > N: \size {x_n - l} < \dfrac \epsilon 2$

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

\(\displaystyle \size {x_n - x_m}\) \(=\) \(\displaystyle \size {x_n - l + l - x_m}\)
\(\displaystyle \) \(\le\) \(\displaystyle \size {x_n - l} + \size {l - x_m}\) Triangle Inequality
\(\displaystyle \) \(<\) \(\displaystyle \frac \epsilon 2 + \frac \epsilon 2\) by choice of $N$
\(\displaystyle \) \(=\) \(\displaystyle \epsilon\)

Thus $\sequence {x_n}$ is a Cauchy sequence.

$\blacksquare$


Proof 2

From Real Number Line is Complete Metric Space, $\R$ under the usual metric is a metric space.

The result then follows as a special case of Convergent Sequence is Cauchy Sequence.

$\blacksquare$


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Sources