The Sudakov's Minoration Inequality states the following:
Let $(X_t)_{t \in T}$ be a mean zero Gaussian process. Then for any $\epsilon \geq 0$, we have $$ \mathbb{E}\sup_{t \in T} X_t \geq c\epsilon \sqrt{\log N(T, d, \epsilon)}, $$ where $d(t,s) := \| X_t - X_s \|_{L^2}$. Here we define $N(T, d, \epsilon)$ as the smallest cardinality of the $\epsilon$-net $\mathcal{N}$ on $T$ using the metric $d$. (That is, $\mathcal{N}$ is an $\epsilon$-net if for all $z \in T$, there exists some $t \in \mathcal{N}$ such that $d(t, z) \leq \epsilon$.)
The proof of the theorem starts off by assuming $N(T, d, \epsilon) < +\infty$ and states that the case $N(T, d, \epsilon) = +\infty$, i.e. we do not admit a finite $\epsilon$-net, is almost immediate as $\mathbb{E}\sup_{t \in T} X_t = \infty$. However, I do not see why it must be the case that $\mathbb{E}\sup_{t \in T} X_t = \infty$. How do we deduce this result?