Functional Analysis: Banach Spaces

Show that $\displaystyle C(\bar{\Omega})$ with norm $\displaystyle ||v||_{C(\bar{\Omega})}=\max\{|v(x)| : x\in\bar{\Omega}\}$ is a Banach space but that $\displaystyle C^1(\bar{\Omega})$ with the same norm is NOT a Banach space.

So for the former I need to show that every Cauchy sequence in $\displaystyle C(\bar{\Omega})$ converges in said norm. So do I just define a random Cauchy sequence in the space, say $\displaystyle \{v_n\}$, so we have that $\displaystyle ||v_m-v_n||\rightarrow 0$ as $\displaystyle m,n\rightarrow\infty$. So now do I just substitute this directly into the norm given and show it must converge in that norm?

Then the latter, how would I go about constructing a Cauchy sequence and then showing it doesn't converge in said norm? (Am I even on the right track?)