Let $X$ be a metric space with the distance function $d: X \times X \rightarrow \mathbb{R}$. For a subset $Y$ of $X$, its diameter is defined as $\delta(Y):=\sup \left\{d\left(y, y^{\prime}\right) \mid y, y^{\prime} \in Y\right\}$.

Show that, if $X$ is compact and $\left\{U_{\lambda}\right\}_{\lambda \in \Lambda}$ is an open covering of $X$, then there exists an $\epsilon>0$ such that every subset $Y \subset X$ with $\delta(Y)<\epsilon$ is contained in some $U_{\lambda}$.