As a step in a proof I've been trying to show that convergence of a sequence implies it must have precisely one accumulation point. This is the definition we use for an accumulation point
Let $S$ be a set of real numbers. A real number is an accumulation point $s_0$ of $S$ if and only if for any $\epsilon > 0$, there exists at least one point $t$ of $S$ such that $0 < |t-s_0| < \epsilon$.
I wish to prove:
Lemma: A convergent sequence has precisely one accumulation point.
My thoughts: Since the limit of the sequence exists we know that the set cannot have 2 accumulation points or more, we will show by contradiction.
If we would have multiple accumulation points the limit does not exist because we can never get arbitrarily close to a single point (within ϵ), because there exist certain subsequences that each get arbitrarily close to at least two accumulation points, which determine the minimum distance a sequence can be from an accumulation point (so we get a lower bound and therefore we do not get convergence). There must be precisely one accumulation point.
I do not know how to make this more precise of a statement, there are a lot of words and it's not really structured. I'm looking for some help making a more rigourous argument.
Best Answer
Suppose the sequence converges to $x$. Clearly $x$ is an accumulation point.
Suppose on the contrary that we have a second accumulation point, $y$. Let $r = \frac{|x-y|}{2}$.
Notice that there exists $N>0$, such that for all $n>N$, then $|x_n -x|<r$.
That is when $n>N$, we have $$|x_n - y|=|x_n-x+x-y| \ge ||x_n-x|-|x-y||=||x_n-x|-2r|=2r-|x_n-x|>r$$
Hence we cannot have infinitely many points that get arbitrarily close to $y$, hence $y$ can't be an accumulation point.