Lebesgue Number Lemma for a Sequentially Compact Subset of a Metric Space

Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$, which is a topology on $X$ by Metric Open Sets Form a Topology. Let $K\subseteq X$ be sequentially compact in $(X,d)$, let $I$ be a set, and let $(U_i)_{i\in I}$ be an open cover of $K$ in $(X,\mathcal{T}_d)$.

Then there exists a real number $\delta>0$ such that for every $x\in K$ there exists $i\in I$ with

$$B_d(x,\delta)\subseteq U_i,$$

where $B_d(x,\delta)$ is the open ball in $X$ with center $x$ and radius $\delta$.