Limit Point of Sequence is Accumulation Point

Theorem

Let $X$ be a topological space.

Let $A \subseteq X$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $A$.

Let $\alpha$ be a limit point of $\left \langle {x_n} \right \rangle$.

Then $\alpha$ is also an accumulation point of $\left \langle {x_n} \right \rangle$.

Proof

Let $\alpha$ be a limit point of $\left \langle {x_n} \right \rangle$.

Then by definition of limit, $\left \langle {x_n} \right \rangle$ 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 $\left \langle {x_n} \right \rangle$.

Hence the result, from definition of accumulation point.

$\blacksquare$

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Limit Points