Real Analysis, can someone help me?

$\displaystyle $$\text{Let }\left( \text{X},\text{ d} \right)\text{ a metric space}.\text{ Show that }{{\left\{ {{\text{x}}_{\text{n}}} \right\}}_{\text{n}\in \mathbb{N}}}~\text{satisfies }{{\text{x}}_{\text{n}}}\to \text{x when n}\to \infty \text{ if and only if every subsequence }\!\!\{\!\!\text{ }{{\text{x}}_{{{n}_{k}}}}{{\text{ }\!\!\}\!\!\text{ }}_{k\in \mathbb{N}}}\subset {{\left\{ {{\text{x}}_{\text{n}}} \right\}}_{\text{n}\in \mathbb{N}}}\text{has a sub}-\text{subsequence}~{{\text{ }\!\!\{\!\!\text{ }{{\text{x}}_{{{n}_{{{k}_{j}}}}}}\text{ }\!\!\}\!\!\text{ }}_{j\in \mathbb{N}}}\subset {{\text{ }\!\!\{\!\!\text{ }{{\text{x}}_{{{n}_{k}}}}\text{ }\!\!\}\!\!\text{ }}_{k\in \mathbb{N}}}\text{ such that }{{\text{x}}_{{{n}_{{{k}_{j}}}}}}\to x\text{ when j}\to \infty $$$

Re: Real Analysis, can someone help me?

If $\displaystyle x_n$ converges, it is obvious that $\displaystyle x_{n_{k_j}}$ converges, right?

Suppose that $\displaystyle x_n$ does not converge to x. What does that mean in terms of N and $\displaystyle \epsilon$? Does that give you an idea for the sequence $\displaystyle x_{n_k}$?

- Hollywood

Re: Real Analysis, can someone help me?

Bastiante study this theorem

Theorem

If a sequence ( Xn ) of real numbers converges to a real number x then any subsequence of X also converges to x.

Proof:

Let ε >0 be given and let K(ε) be such that if n>= K(ε) , then | xn - x|<ε . If r1<r2<r3…<rn <… is an increasing sequence of natural numbers, it can easily be proven by induction that rn>=n . Hence , if n >=K(ε) we also have rn>=n>=K(ε) so that |xrn – x| <ε . This proves that the subsequence ( Xrn ) also converges to x , and the theorem is proved.

MINOAS