Proving Corollary 1.5 in Katznelson's Intro to Harmonic Analysis

Question

Corollary 1.5 in Katznelson's Intro to Harmonic Analysis states: Assume $f_j \in L^1(\mathbb{T}), j = 0, 1, 2, ...,$ and $||f_j - f_0||_{L^1} \rightarrow 0$ Then $\hat{f}(n) \rightarrow \hat{f}_0 (n)$ uniformly.

where $L^1(\mathbb{T})$ denotes the space of all complex valued Lebesgue integrable functions on $\mathbb{T} = \mathbb{R}/2 \pi \mathbb{Z}$. How would I go about proving this?

EDIT: This is a corollary to theorem 1.4 which states:

1. $\hat{(f + g)} (n) = \hat{f}(n) + \hat{g}(n)$
2. For any complex number $\alpha$, $\hat{(\alpha f)}(n) = \alpha \hat{f}(n)$
3. If $\bar{f}$ is the complex conjugate of $f$, then $\hat{\bar{f}} = \bar{\hat{f}(-n)}$.
4. $\hat{f}_{\tau} (n) =\hat{f}(n)e^{-in\tau}$, where $f_{\tau}(t) = f(t - \tau)$.
5. $|\hat{f}(n)| \leq \frac{1}{2\pi} \int |f(t)| dt= ||f||_{L^1}$

Answer 1

Given $\varepsilon>0$, there exists $K\in\mathbb{N}$ with $\|f_k-f_0\|<\varepsilon$ for all $k\geq K$. Then by $(5)$ we have $$|\hat{f}_k(n)-\hat{f}_0(n)|\leq\|f_k-f_0\|<\varepsilon $$ for all $n\in\mathbb{Z}$ and $k\geq K$, hence $\hat{f}_k\to\hat{f}_0$ uniformly.