1. ## Converging Subsequence

Suppose that a sequence {$\displaystyle x_n$} is monotone and that it has a convergent subsequence. Show that {$\displaystyle x_n$} also converges.

I think that "since the subsequence is convergent, the sequence {$\displaystyle x_n$} is bounded", and
according to the monotone convergence theorem,a monotone sequence is convergent if and only if it is bounded.

I am not sure about the first part of my reasoning, because there is a theorem that says "every bounded sequence has a convergent subsequence". Based on that, I am stating that "since the subsequence is convergent, the sequence is bounded".

Is this the correct reasoning or are there any other ways this statement can be proved? Suggestions will be appreciated

2. Originally Posted by harish21
Suppose that a sequence {$\displaystyle x_n$} is monotone and that it has a convergent subsequence. Show that {$\displaystyle x_n$} also converges.

I think that "since the subsequence is convergent, the sequence {$\displaystyle x_n$} is bounded", and
according to the monotone convergence theorem,a monotone sequence is convergent if and only if it is bounded.

I am not sure about the first part of my reasoning, because there is a theorem that says "every bounded sequence has a convergent subsequence". Based on that, I am stating that "since the subsequence is convergent, the sequence is bounded".

Is this the correct reasoning or are there any other ways this statement can be proved? Suggestions will be appreciated
Assume WLOG that $\displaystyle x_n$ is increasing. We know that there exists some $\displaystyle \{x_{\varphi(n)}\}$ such that $\displaystyle x_{\varphi(n)}\to x$. So, let $\displaystyle \varepsilon>0$ be given. Then, there exists some $\displaystyle N\in\mathbb{N}$ such that $\displaystyle N\leqslant n\implies \left|x-x_{\varphi(n)}\right|=x-x_{\varphi(n)}<\varepsilon$. But, if $\displaystyle \varphi\left(N\right)\leqslant n\implies x_{\varphi(N)}\leqslant x_n\implies x-x_n\leqslant x-x_{\varphi(N)}<\varepsilon$