Prove if $ \lim_{n \rightarrow \infty} {x_{nk}} = l$ for some $l \in R$, then$\displaystyle \lim_{n \rightarrow \infty} {x_{n}} = l$

Question

Hi folks I know that I should probably be using the Bolzano-Weierstrass theorem and the definition of a Cauchy sequence, but I am not really sure where to get started.

Let $(x_n)$ be Cauchy Sequence and let $(x_{nk})$ be a subsequence such that $\displaystyle \lim_{n \rightarrow \infty} {x_{nk}} = l$ for some $l \in R$, prove $\displaystyle \lim_{n \rightarrow \infty} {x_{n}} = l$.

Answer 1

Let $\varepsilon > 0$ be given, there is $N_1 > 0$ such that: $|x_n - x_m| < \varepsilon/2$ when $m, n > N_1$. Since $n > N_1$ $\Rightarrow$ $n_k > N_1$ $\Rightarrow$ $|x_n - x_{n_k}| < \varepsilon/2$ . Also there is $N_2 > 0$ such that $|x_{n_k} - L| < \varepsilon/2$ if $n > N_2$. Choose $N = \max\{N_1, N_2\}$ then if $n > N$ $\Rightarrow$ $|x(n) - L| < |x_n - x_{n_k}| + |x_{n_k} - L| < \varepsilon/2 + \varepsilon/2 = \varepsilon$.