general question about polynomials converging to a function

Question

I have some function $f$. Then I have a sequence of polynomials $P_n$ that converges uniformly to $f$ on an interval $I=[0,1]$. Now does the same sequence of polynomials converge uniformly on interval $D=(0,1] \subset I$? If so can you point me towards some proof of this or some logic behind it?

Answer 1

Yes, it is true. Take $\varepsilon>0$. You are assuming that there is a natural $N$ such that$$(\forall n\in\mathbb{N})(\forall x\in[0,1]):n\geqslant N\implies\bigl|f(x)-P_n(x)\bigr|<\varepsilon.$$But if this is true, then, in particular,$$(\forall n\in\mathbb{N})(\forall x\in(0,1]):n\geqslant N\implies\bigl|f(x)-P_n(x)\bigr|<\varepsilon,$$since $(0,1]\subset[0,1]$.