# Help undestanding compactness with convergent subsequences

One way to define compactness in metric spaces is to note that in compact metric space each sequence has a convergent subsequence.

Understanding compactness is difficult for me from this perspective. Let's take the set of real numbers as an example. Say my sequence is something like $(1,2,3,...)$ that isn't convergent. But why can't I pick, say the subsequence $(1,1,1,...)$, i.e. a sequence that is constant, and so convergent? And similarly for every possible sequence in $\mathbb{R}$?

$(1,1,1,\dots)$ is not a subsequence of $(1,2,3,\dots)$.
If you have a sequence $a_1,a_2,\dots$, then a subsequence of that is any sequence $$a_{i_1},a_{i_2},a_{i_3},\dots$$ where $i_1,i_2,i_3,\dots$ are some natural numbers such that $$i_1<i_2<i_3<\dots.$$
This means that for the sequence $(1,2,3,\dots)$, you can take $i_1=1$ so the first element of the subsequence will be $1$, but you cannot take $i_2=1$ because $i_2$ must be larger than $i_1$.