Problem: If $(f(t),\phi)$ is $C^1$ for all $\phi\in B^*$, then it is $C^0$, where $f(t)$ is Banach space valued function from closed real interval.
Proof: By first use of uniform boundedness (It appears this is used in the second use?) $$\|f(t)\|<N$$ And by second use of uniform boundedness (note $\lim_{h\rightarrow \infty}\left(\frac{f(t+h)-f(t)}{h},\phi\right)$ exists):
$$\left\|\frac{f(t+h)-f(t)}{h}\right\|<M$$ And now the statement follows immediately as $\|f(t+h)-f(t)\|<Mh$, so that taking the limit gives zero.
Question: How was the uniform boundedness used in above, I have some trouble seeing both results?