Limit Point of Sequence is Accumulation Point

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.


Let $\sequence {x_n}$ be a sequence in $A$.

Let $\alpha$ be a limit point of $\sequence {x_n}$.


Then $\alpha$ is also an accumulation point of $\sequence {x_n}$.


Proof

Let $\alpha$ be a limit point of $\sequence {x_n}$.

Then by definition of limit, $\sequence {x_n}$ converges to $\alpha$.

By definition of convergence that means:

for any open set $U \subseteq T$ such that $\alpha \in U$: $\exists N \in \R: n > N \implies x_n \in U$.

As there is an infinite number of values of $n > N$, there are an infinite number of terms of $\sequence {x_n}$.

Hence the result, from definition of accumulation point.

$\blacksquare$


Sources