Accumulation point in real spaces

Question

Sequences in $\mathbb{R}^n$ have a unique limit. Is it true that for any sequence which converges to limit exist there exists no accumulation point a such that $x \neq a$. i.e. does unique limit ensures unique accumalation point. Can this be generalized to metric spaces.

Answer 1

Yes, this holds in any metric space. If $(x_n)_n \rightarrow x$ and $p$ is an accumulation point of $(x_n)$, then there is a subsequence $(x_{n_k})_k$ such that $(x_{n_k})_k \rightarrow p$.

But any subsequence of a convergent sequence converges to the same limit, so $(x_{n_k})_k \rightarrow x$. So by unicity of limits (applied to the subsequence) we have $x = p$. So the only accumulation point is $x$ (and a limit is always an accumulation point of a sequence).