Let $(a_n)_n$ be a sequence in $\mathbb R$ and $A := \{a_n \,|\, n \in \mathbb N\}$ the set of elements of the sequence. I want to show
If $(a_n)_n$ is injective, then a limit point of $(a_n)_n$ is also a cluster point of $A$.
My definitions are
$x_0$ is a cluster point of a set $A \iff$ for every $\varepsilon \gt 0$ there exists $a \in A$, such that $0 \lt |a - x_0| \lt \varepsilon$.
$x_0$ is a limit point of a sequence $(a_n)_n \iff$ there is a subsequence which converges to $x_0$ or equivalently if for all $\varepsilon \gt 0$ and all $N \in \mathbb N$ there exists $n \ge N$, such that $|a_n - x_0| \lt \varepsilon$.
I don't know how to start. How should I use the injective property?