Show that with norm is a Banach space but that with the same norm is NOT a Banach space.

So for the former I need to show that every Cauchy sequence in converges in said norm. So do I just define a random Cauchy sequence in the space, say , so we have that as . 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?)