Convergence in norm in $\ell^1$ space

Question

For a sequence $\{x_n\}\in \ell^1$, if $x_n(k)\rightarrow x(k)$ for all $k$, it is in general not true that $\|x_n-x\|_1\rightarrow 0$. But if $x_n(k)$ are Fourier series of a function $f_n(z)$ and $f_n(z)$ converges to $f(z)$ uniformly and $f_n(z)$ are uniformly bounded, in this case, $\|x_n-x\|_1\rightarrow 0$ is true?