Bolzano-Weierstrass Theorem in Euclidean Space

Let $n$ be a natural number and let $\mathbb{R}^n$ be Euclidean space equipped with the Euclidean distance $d_E$, which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$. Let $K\subseteq\mathbb{R}^n$ be bounded in $(\mathbb{R}^n,d_E)$, and let $(x^{(m)})_{m\in\mathbb{N}}$ be a sequence in $\mathbb{R}^n$ with $x^{(m)}\in K$ for every $m\in\mathbb{N}$.

Then there exist a point $\ell\in\mathbb{R}^n$ and a strictly increasing sequence $(p_k)_{k\in\mathbb{N}}$ in $\mathbb{N}$ such that the subsequence $(x^{(p_k)})_{k\in\mathbb{N}}$ converges to $\ell$ in $(\mathbb{R}^n,d_E)$.