Let V be the vector space of continuously differentiable functions on [a, b] with the norm $$\|f\|=\max _{[a, b]}|f(t)|+\max _{[a, b]}\left|f^{\prime}(t)\right|$$ Show that V is complete.
I was trying to do so by taking an $f_n(t)$ and assuming it cauchy sequence, hence:
for $n,m>n_0$ $$max|f_n(t)-f_m(t)|<\epsilon$$ Now I just need to show that $f_n$ converge in the given norm. $$\|f_n(t)-f_m(t)\|=\max _{[a, b]}|f_n(t)-f_m(t)|+\max _{[a, b]}|f'_n(t)-f'_m(t)|<\epsilon + \max _{[a, b]}|f'_n(t)-f'_m(t)|$$
How do I show that $$\max _{[a, b]}|f'_n(t)-f'_m(t)|<\epsilon$$