Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 2
Theorem
Let $\sequence {x_n}$ be a sequence in $\R$.
Let $\sequence {x_n}$ be a Cauchy sequence.
Then $\sequence {x_n}$ is convergent.
Proof
Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
By Real Cauchy Sequence is Bounded, $\sequence {a_n}$ is bounded.
By the Bolzano-Weierstrass Theorem, $\sequence {a_n}$ has a convergent subsequence $\sequence {a_{n_r} }$.
Let $a_{n_r} \to l$ as $r \to \infty$.
It is to be shown that $a_n \to l$ as $n \to \infty$.
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.
Then $\dfrac \epsilon 2 > 0$.
Hence:
- $(1): \quad \exists R \in \R: \forall r > R: \size {a_{n_r} - l} < \dfrac \epsilon 2$
We have that $\sequence {a_n}$ is a Cauchy sequence.
Hence:
- $(2): \quad \exists N \in \R: \forall m > N, n > N: \size {x_m - x_n} \le \dfrac \epsilon 2$
Let $n > N$.
Let $r \in \N$ be sufficiently large that:
- $n_r > N$
and:
- $r > R$
Then $(1)$ is satisfied, and $(2)$ is satisfied with $m = n_r$.
So:
\(\ds \forall n > N: \, \) | \(\ds \size {a_n - l}\) | \(=\) | \(\ds \size {a_n - a_{n_r} + a_{n_r} - l}\) | |||||||||||
\(\ds \) | \(\le\) | \(\ds \size {a_n - a_{n_r} } + \size{a_{n_r} - l}\) | Triangle Inequality for Real Numbers | |||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac \epsilon 2 + \dfrac \epsilon 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
So, given $\epsilon > 0$, we have found $n \in \R$ such that:
- $\forall n > N: \size {a_n - l} < \epsilon$
Thus:
- $x_n \to l$ as $n \to \infty$.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 5$: Subsequences: Cauchy sequences: $\S 5.19$
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces