I'm going through some problems and I'm really stumped on this one. The questions says that
Given $f(x)=|x|$, show that there is a sequence of (real) polynomials $P_n(x)$ with $P_n(0)=0$ that converge uniformly to $f(x)$ on the interval $[-1,1]$.
I think an application of the Weierstrass theorem is in order, but I don't know how to apply it here and so I'll need some help.
Thanks.
Best Answer
As you correctly guessed, you can use that by Weierstrass's theorem there is a sequence of polynomials $Q_n(x)$ uniformly converging to $f$ on $[-1,+1]$.
Done? Not quite: those $Q_n$'s might not satisfy $Q_n(0)=0$.
Well, then give them a little push that will force them to comply: do you see what you have to add to each of them to obtain $P_n$ and why the push is little (and becomes littler and littler) ?
And do you see why the new sequence of $P_n$'s will still converge to $f$ uniformly because of the mentioned littleness?
And do you see that the exact formula for $f$ is a red herring and that only the fact that $f(0)=0$ is relevant?
Yes, I'm sure you'll see all that after a short moment a reflexion. Good luck!