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I found in my professor's lecture note that $\mathop {\lim }\limits_{k \to \infty } {x_k} = x$ is equivalent to "$\exists x$ such that every subsequence $x_{k_l}$ has a subsequence $x_{k_{l_t}} \rightarrow x$".

However, I am puzzled why it is not simply "$\exists x$ such that every subsequence $x_{k_l} \rightarrow x$"? Is my professor right? If my professor is correct, why "$\exists x$ such that every subsequence $x_{k_l} \rightarrow x$" does not lead to $\mathop {\lim }\limits_{k \to \infty } {x_k} = x$? Thank you.

Tony
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1 Answers1

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He is right, there is a proposition that says;

$x_n \to x$ if and only if for every $(x_{n_{k}})$ subsequence of $(x_n)$ you have that $x_{n_k} \to x$ when $k \to \infty$

There is another proposition that says that

$x_n \to x$ if for every subsequence $(x_{n_k})$ of $(x_n)$ there exist a subsequence $(x_{n_{k_j}})$ of $(x_{n_k})$ such that $x_{n_{k_j}} \to x$

There are two different equivalences of $x_n \to x$. You may find useful any of them when trying to solve exercises. Its a nice exercise to prove the second statement if your professor hasn't done it yet.

Joaquin Liniado
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