Showing two norms are not equivalent

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I am struggling with part d of this question. So the uniform norm is just the supremum ( f(t) ). I tried going for the definition of equivalent norms which I can't seem to find constants to bound each of the norms.

My thought is since the vector space is complete with respect to the uniform norm I could go down the route of showing it is not complete with the a norm(defined in part b). So I would need to show that there's a cauchy sequence that doesn't converge in the space, I am unsure what counterexample I could think of if anybody could point me in a direction to start?

It would be helpful to understand what points I should be looking at and the thought process to constructing a counterexample for such a space

I understand why we want that but I'm terrible at coming up with an example.. I think the first thing I should do is think of a function so that points $f_n$ with $|f_n|_a$ goes to 0 but $|f_n|_{\infty}$ that doesn't, so I want to essentially fix a value for the uniform norm but let the first one tend to zero?

The solution said 1-tn for 0 <= t <= 1/n
and 0 otherwise, so I'm trying to look past that

would e^-n work? Since the uniform norm is 1 and the a norm is 0 at this same point

Sorry,

e^-nx is what I meant, I think this works?