# How to show unit closed ball of $C^1[0,1]$ is totally bounded using definition?

I had shown that unit ball of $$C^1[0,1]$$ is compact with norm $$||f||_1=||f||_{\infty}+||f'||_{\infty}$$ using Arzela Ascoli thoerem.

Compactness implies Totally bounded.

Is there is any other way to directly prove this fact that unit closed ball is totally bounded?

Your space is an infinite dimensional normed linear space. The unit ball of such a space is never compact. What Arzela - Ascoli theorem tells you is that if $$(f_n)$$ is in the unit ball of this space then there is a subsequence which converges in the sup norm. But this does not give convergence in the norm of $$C^{1}[0,1]$$.