Proof of uniform convergence if VC dimension is finite
Let $X = |L_\mathcal{D}(h) - L_S(h)|$. The statement on the expectation of the supremum of $X$ implies, in particular, that for some $M$,
$$
\mathbb{E}[X] \leq M.
$$
Since $X \geq 0$, Markov's inequality implies that
$$
\Pr\left[X \geq \frac{M}{\delta}\right] \leq \delta.
$$
This implies that
$$
\Pr\left[X \leq \frac{M}{\delta}\right] \geq
\Pr\left[X &lt; \frac{M}{\delta}\right] =
1 - \Pr\left[X \geq \frac{M}{\delta}\right] \geq 1 - \delta.
$$