A Cluster Point of a Sequence in a Metric Space is the Limit of a Subsequence

Let $(X,d)$ be a metric space, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$, and let $x\in X$ be a cluster point of $(x_m)_{m\in\mathbb{N}}$ in $(X,d)$. Let $(\varepsilon_k)_{k\in\mathbb{N}}$ be a sequence in the real numbers with $0<\varepsilon_k$ for every $k\in\mathbb{N}$.

1. There is a strictly increasing sequence $(n_k)_{k\in\mathbb{N}}$ in $\mathbb{N}$, in the sense of Subsequence of a Sequence in a Set, such that

$$d(x_{n_k},x)<\varepsilon_k\qquad\text{for every }k\in\mathbb{N}.$$

2. If moreover $(\varepsilon_k)_{k\in\mathbb{N}}$ has limit $0$, then for any such $(n_k)_{k\in\mathbb{N}}$ the subsequence $(x_{n_k})_{k\in\mathbb{N}}$ converges to $x$ in $(X,d)$.