About uniform convergence in Hille-Yosida's Theorem proof

Question

In every proof of H-Y's Theorem (the sufficiency part) that I saw, some form of uniform convergence is mentioned, in a very terse way.
It goes like this, following Pazy's book: since, for every $x$ in a Banach space $X$ and $\lambda,\,\mu > 0$, one has $$\left\| \exp(t A_\lambda)(x) - \exp(t A_\mu)(x)\right\| \leqslant t\left\|A_\lambda(x) - A_\mu(x) \right\|, $$ for $t \geqslant 0$, in which $A_\lambda$ is the Yosida approximation of a linear operator $A$ satisfying the hipothesis of H-Y's Theorem (this is Lemma 3.4 in Pazy's book) and $A_\lambda(x) \longrightarrow A(x)$, as $\lambda \longrightarrow \infty$ (this is Lemma 3.3 in Pazy's book), for $x$ in the domain of $A$, Pazy then writes that $(\exp(t A_\lambda)(x))_{\lambda}$ converges uniformly on bounded intervals of the form $[0,\,t_0]$.
That said, I'm struggling to understand what kind of uniform convergence is this, since, a priori, is not related to the strong operator topology at all. That put, how can I understand this statement?
Since we're talking about taking $\lambda \longrightarrow \infty$, where $\lambda$ is any positive real number, this looks like it is related to the concept of a net, which I have very little acquaintance with, or it could be much simpler than that.
Any help is appreciated.