In his famous book, just before proving Hille-Yosida theorem, Brezis gives a generalization of the Cauchy-Lipschitz theorem for existence and unicity in Banach spaces, which I will call $E$. At the beginning he states that the space $$X= \left\lbrace u \in C\bigl([0, \infty);E\bigr) \mid \sup_{t \geq 0} e^{-kt} \|u(t)\| < \infty \right\rbrace$$ is a Banach space for every fixed $k$.
Now, I take a Cauchy sequence in $X$ which I call $\{u_n\}$. Then $\forall \epsilon >0$ it exists $N \in \mathbb{N}$ such that $\|u_n-u_m\|_X < \epsilon$ if $n,m \geq N$. Then I can prove that for every fixed $t_0$ in $[0,\infty)$ I have that $\{u_n(t_0)\}$ is a Cauchy sequence in $E$, because I can just take $\epsilon '=\epsilon e^{-kt_0}$ and then use some trivial estimates to prove that $\|u_n(t_0) - u_m(t_0)\| < \epsilon$ for $n,m$ big enough. The problem is that if I now define $u(t)= \lim_{n\to \infty} u_n(t)$ I don't know how to prove it is continuous. Indeed I would estimate $\|u(t)-u(t_0)\|$ with $$\|u(t)-u_n(t)\|+\|u_n(t)-u_n(t_0)\|+\|u_n(t_0)-u(t_0)\|$$ But from here I don't know how to continue, because I would need for instamce that $\forall n>M$ $\|u_n(t)-u_n(t_0)\|\leq \epsilon$, but with only punctual convergence I should not be able to conclude. Maybe it is very simple and I am missing some trivial fact, but I don't see it right now.