Hello everyone. I am posting this not because the normal proof is difficult, but because I think I have come up with an alternative way of doing it and am wondering if I made any fatal errors. Also, half the reason I am putting this up here is to improve my often remarked upon lack of coherency. So any constructive criticism in either category is much appreciated.

Definitions: If $\displaystyle \chi$ is a metric space and $\displaystyle \left\{p_n\right\}$ is a sequence under $\displaystyle \chi$, then we define an infinite subsequence of $\displaystyle \left\{p_n\right\}$ as follows: If we have a infinite sequence of natural numbers $\displaystyle n_1,n_2,\cdots,n_k$ where $\displaystyle n_1<n_2<\cdots<n_k$ then the sequence $\displaystyle \left\{p_{n_k}\right\}$ is a subsequence of $\displaystyle \left\{p_n\right\}$.

So now that we have defined an infinite subsequence, let's get to the question.

Question: If $\displaystyle \left\{p_n\right\}$ is a convergent sequence under $\displaystyle \chi$ which converges to $\displaystyle p$ then prove that every infinite subsequence $\displaystyle \left\{p_{n_k}\right\}$ of $\displaystyle \left\{p_n\right\}$ converges to $\displaystyle p$.

Answer: I think I know how they intended it to be done, but I came up with a different method and would appreciate any crticism.

1. In the definition of the subsequence we defined the $\displaystyle n_k$ in $\displaystyle \left\{p_{n_k}\right\}$ to be sequence of natural numbers such that $\displaystyle n_1<n_2<\cdots<n_k$. So instead of $\displaystyle n_k$ let us define it as follows: let $\displaystyle n_k$ be represented by $\displaystyle \varphi: \mathbb{N}\mapsto\mathbb{N}$ which is a function of $\displaystyle n$ and has the quality that it is monotonically increasing, so then we may represent a subsequence of $\displaystyle \left\{p_n\right\}$ as $\displaystyle p_{\varphi}$.

2. We know because $\displaystyle \left\{p_n\right\}$ converges that for every $\displaystyle \varepsilon>0$ there exists and integer $\displaystyle N$ such that $\displaystyle \forall{N\leqslant{n}}~d_{\chi}\left(p,p_n\right)< \varepsilon$.

3. So from this we can make the observation that $\displaystyle d_{\chi}\left(p,p_{f}\right)<\varepsilon$ whenever $\displaystyle N\leqslant{f^{}\left(n\right)}\implies{\lceil{\til de{\varphi}\left(N\right)\rceil}}=N_0\leqslant{n}$. Where $\displaystyle \tilde{\varphi}$ is the generalized inverse function.

4. Stating the end part of 4. more explicitly: for every $\displaystyle \varepsilon>0$ there exists an integer $\displaystyle N_0$ such that $\displaystyle d_{\chi}\left(p,p_f\right)<\varepsilon$ whenever $\displaystyle N_0\leqslant{n}$. This the definition of a sequence which converges to the value $\displaystyle p$ which is what was asked to be proven.

Any comments greatly appreciated,

Mathstud.