# Question about Hille Yosida Theorem proof

I'm working on Hille-Yosida theorem on Vrabie's book. Here is the statement:

In order to prove the sufficiency two lemmas are needed:

and

Here comes my question, is highlighted in yellow:

Why can we deduce that there exists such operator $$S(t)$$? How can we justify that such limit exist? Because of boundedness? Once we know that there exits is easy to show that $$\|e^{tA_\lambda}x-S(t)x\|\to 0$$ since we have $$(3.2.5)$$ and $$(3.2.3)$$.

On the other hand, about the words that are highlighted in blue, the convergence in norm implies uniform convergence, but can we choose that convergence just for compact subsets of $$R_+$$?.

In fact, the convergence for any $$x\in D(A)$$ follows, as said, from (3.2.5), for $$0\leqslant a, since we have
$$\underset{t\in[a,b]}{\sup}\lVert e^{tA_\lambda} x - e^{tA_\mu} x\rVert \leqslant b \lVert A_\lambda x - A_\mu x\rVert \underset{\lambda,\mu \rightarrow \infty}{\longrightarrow} 0.$$
Above limit is true since for all $$x\in D(A)$$, the following convergence holds in $$X$$ (see (3.2.3)),
$$A_\lambda x \underset{\lambda \rightarrow \infty}{\longrightarrow} Ax .$$
Thus, $$(e^{tA_\lambda} x)_{\lambda>0}$$ is a Cauchy net in $$X$$ for all $$x\in D(A)$$, hence admits a limit in $$X$$ denoted by $$S(t)x$$.