Accumulation Point of Sequence is not necessarily Limit Point

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

Let $\sequence {a_n}$ be a sequence in $T$.

Let $q \in S$ be an accumulation point of $\sequence {a_n}$.

Then it is not necessarily the case that $q$ is also a limit of $\sequence {a_n}$.

Proof
Proof by Counterexample:

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\sequence {a_n}$ be the sequence defined as:

Then $\sequence {a_n}$ has exactly one accumulation point, that is $1$.

However, $1$ is not a limit of $\sequence {a_n}$, as $\sequence {a_n}$ has no limit.

Hence the result.