Let $(g_n)$ be a sequence of functions on $[a,b]$. If $(g_n)$ converges pointwise and uniformly continuous on $[a,b]$, does $(g_n)$ also converge uniformly on $[a,b]$?


Take $g_n(x)=x^n;x\in [0,1]$

Every continuous function on a compact set is uniformly continuous

But $\lim_{n\to \infty} g_n(x)=g(x)=0 ;0\le x<1 \text{and} 1,x=1$

which is not even continuous

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