(Background definitions)
$A(\mathbb{T})$ is the space of all $2\pi$ periodic functions $f$ such that $\sum\limits_{k=-\infty}^{\infty}|\widehat{f}(k)| < \infty$. It is a normed space when normed by $\|\cdot\|_{A(\mathbb{T})} = \sum\limits_{k=-\infty}^{\infty}|\widehat{f}(k)|$.
Also, $\widehat{f}(k) = \int\limits_{0}^{2\pi}f(t)e^{-ikt}dt$.
I have a homework problem which says the following:
Let $f_{n}\in A(\mathbb{T})$ for each $n\geq 1$, and let $\|f_{n}\|_{A(\mathbb{T})}\leq 1$ for each $n\geq 1$. Assume $f_{n}\to f$ in $\|\cdot\|_{\infty}$.
a) Show that $f\in A(\mathbb{T})$ and $||f||_{A(\mathbb{T})}\leq 1$.
b) Show that the given hypothesis are not enough to guarantee that $\|f_{n} - f\| _{A(\mathbb{T})}\rightarrow 0$.
My progress:
I have solved the first part by showing that each partial sum $\sum\limits_{k=-n}^{n}|\widehat{f}(k)|\leq 1 + \epsilon$ for every $\epsilon > 0$.
But this has given me little insight for the second problem. Every example I come up with satisfies $\|f_{n} - f\| _{A(\mathbb{T})}\rightarrow 0$. Even as I am writing this it has just occured to me that I've only considered continuous functions. So I will try some discontinuous ones next but so far this question has completely eluded me.
Any suggestions of what type of functions to focus on?
If I take $f_{n} = \frac{1}{n}$, then $f_{n}\rightarrow 0$ in $\|\cdot\|_{\infty}$ and $\|\cdot\|_{A(\mathbb{T})}$.
If I take $f_{n} = \frac{(-1)^{n}}{n}$, then then $f_{n}\rightarrow 0$ in $\|\cdot\|_{\infty}$ and $\|\cdot\|_{A(\mathbb{T})}$.
If I take $f_{n} = \sum\limits_{j=1}^{n}\frac{(-1)^{j}}{j}$, then $f_{n}\rightarrow 0$ in $\|\cdot\|_{\infty}$ and $\|\cdot\|_{A(\mathbb{T})}$.
If I take $f_{n} = \sum\limits_{j=1}^{n}\frac{1}{j}$, then $f_{n}$ diverges in any norms.
In fact I think that a sequence of constant functions converges to $f$ in $\|\cdot\|_{\infty}$ if and only if it converges to $f$ in $\|\cdot\|_{A(\mathbb{T})}$.
So in conclusion, I am back where I started.
NOTE: I had asked this question previously but didn't get any real answer yet. But since the question was technically "answered", there were no more views. So I deleted/reposted. I hope this is OK.