Why does this inequality imply non-separability?

In [Da Prato, Giuseppe, and Jerzy Zabczyk, Stochastic equations in infinite dimensions. Cambridge University Press, 1992] p. 23 the authors give the following explanation of why the space $L(U,H)$ of linear operators between infinite dimension separable Hilbert spaces is not separable.

Let $H=L^2(\mathbb{R})$, and define for $t\in\mathbb{R}_{+}$ the isometry $S(t):H\rightarrow H$ by $$S(t)x = x(z+t),\;\text{for}\;x\in H, z\in\mathbb{R}$$ Assuming that $t>s$, $x\in H$ then $$|(S(t) - S(s))x| = |S(s)(S(t-s)x - x)| = |S(t-s)x - x|$$ If we suppose the support of $x$ is in the interval $\left]-\frac{t-s}{2}, \frac{t-s}{2}\right[$ then the functions $S(t-s)x$ and $x$ have disjoint supports. Hence $$|(S(t) - S(s))x|^2 = 2|x|^2$$ and thus $$|S(t) - S(s)| > \sqrt{2}$$ from here we conclude that $L(H,H)$ is not separable.

What is the concluding argument here?

Since $\|S(t)-S(s)\|>\sqrt2$, the balls $B_{\sqrt2/2}\bigl(S(t)\bigr)$ and $B_{\sqrt2/2}\bigl(S(s)\bigr)$ do not intersect. Since there are uncountably many such balls, no countable set can have at least one element in each of them. So, no countable set is dense.

The authors are using the following proposition:

In a metric space $X$, if you can find an uncountable set $U \subseteq X$ such that $\inf_{x,y \in U}d(x,y) > 0$, then the metric space is not separable.