Find a sequence of Lipschitz continuous functions on $[0,1]$ whose uniform limit is $\sqrt{x}$.

Question

Find a sequence of Lipschitz continuous functions on $[0,1]$ whose uniform limit is $\sqrt{x}$, which is a non-Lipschitz function.

Answer 1

$f_n:[0,1]\to \mathbb{R}$ defined by $f_n(x)=\sqrt{x+\frac{1}{n}}$ for all $x\in [0,1]$ and for all $n\in \mathbb{N}$.