Let $(f_n)_{n\in\mathbb N}\in (\mathbb C^{[a,b]})^{\mathbb N}$ which validates the property :
$\forall \varepsilon >0, \exists\delta>0, \forall x,y\in[a,b], |x-y|\le\delta \implies \forall n\in \mathbb N, |f_n(x)-f_n(y)|\le \varepsilon$
If $(f_n)_n$ converge pointwise, will $(f_n)_n$ converge uniformly ?
It seems to be working. Though I don't know how to begin.