Question about compact sets

Question

I know what compact sets are in $R^n$:A set S of real numbers is called compact if every sequence in S has a subsequence that converges to an element again contained in S.like i know similar definition is in every metric space(please correct me if i am wrong).my question is:every set in arbitrary metric space is compact, because i can take the sequence of elements in that set, and as a subsequence i can take only first element.Hence subsequence contains only one element so it is convergent subsequence, therefore every set in arbitrary metric space is compact.What is wrong in my logic?

Answer 1

What is wrong is that, no, you cannot just take the first element. A subsequence of a sequence $(x_n)_{n\in\Bbb N}$ is a sequence $(x_{n_k})_{k\in\Bbb N}$, where $(n_k)_{k\in\Bbb N}$ is a strictly increasing sequence of natural numbers. And the sequence $1,1,1,1,\ldots$ is not strictly increasing.

Answer 2

What's wrong in your logic is your definition of subsequence. Note that with the way you think about subsequences, all of $\mathbb R^n$ and every one of its subsets would be "compact".