We can construct a sequence such that $\|x_n-x_m\|\gt 1/2$ via using Riesz's Lemma. It's not Cauchy sequence and thus it's not a convergent sequence.
My question : In my notes "since the sequence is not convergent, it doesn't have a convergent subsequence" has been written. But when $(-1)^n$ is not convergent, its subsequence is $(-1)^{2n}$ is convergent to $1$. How can we say it doesn't have a convergent subsequence? Where am I wrong, I couldn't realize. Thanks a lot
Yes, that looks like an error in the notes.
The conclusion does hold, but for different reasons. The fact $\|x_n-x_m\|>\frac 12$ means that no subsequence can be Cauchy, and since convergent sequences are always Cauchy, this means that no subsequence is convergent either.