In section 9 of Terence Tao's book, we have the Heine Borel Theorem:-
If $X \subset \mathbb R$, it is bounded and closed iff any sequence of elements in $X$ has a convergent subsequence with limit in $X$
Everything's fine, it has a nice similarity with Bolzano - Weirstraß, but i cannot understand why it has to be closed?
My attempt to prove (the tougher implication) is like this-
Let $X \subset [-M,M]$.Then the sequence $(x_n)_{n=0}^{\infty}$ containing each element of $X$ is bounded, and letting $L^+$ denote the $\limsup (a_n)_{n=0}^{\infty}$ we have $|L^+|<M$, implying the limit superior is finite and thus it has a convergent subsequence with limit $L^+$.
It seems fine to me, but i can't understand why the condition "$X$ is closed" is necessary.
Thanks for helping :)
Yes, it has a convergent subsequence. And the limit of that subsequence belong to $X$, since $X$ is closed. Otherwise, you cannot assure that that will happen.