Is $x^n$ uniformly convergent on $[0,1)$?

Question

I know that it's not uniformly convergent on $[0,1]$ but on $[0,a]$ with $a < 1$. Does that mean it converges uniformly on $[0,1)$?

Answer 1

A sequence uniformly convergent on $[0,1)$ will also be uniformly convergent on $[0,1]$ because only one point is being added. Assuming of course that it does converge at $1$.

Answer 2

The sequence $f_n(x):=x^n$ converges pointwise to $f(x):=0$ for all $x\in[0,1)$. But plugging $x=1-\frac1n$ into $f_n$, we can certainly say $$ \textstyle \sup_x|f_n(x)-f(x)|\ge(1-\frac1n)^n\tag1 $$ The RHS of (1) tends to a nonzero limit (namely $1/e$), so the LHS does not converge to zero.