Yes $ 0 notin (a, b) $, show that each function continues $ f $ in $ (a, b) $ is the uniform limit of a sequence of polynomials $ (q_ {n}) $, where $ q_ {n} (x) = x ^ {n} p_ {n} (x) $ for polynomials $ p_ {n} $.

A solution for this question was presented (Show that each function continues in a closed interval is the uniform limit of a sequence of polynomials), but I worked with my teacher and he provided me with a different solution, but I am trying to understand the thinking behind mainly The initial steps.

**Solution:**

1) Define: $$ F (x) = Bigg { begin {array} {11}

f (x), y x in (a, b) \

frac {f (a)} {a} (x-a) + f (a) & x in (0, a)

end {array} $$

2) According to the Weirstrauss approximation theorem (not Stone – Weirstrauss, but the simplest version). There is a polynomial $ p_ {n} (x) a F (x) $ evenly in $ (0, b) $.

3) Define polynomial $ q_ {n} (x) = p_ {n} (x) – p_ {n} (0) $. Note: $ p_ {n} (0) a 0 = f (0). $

4) Therefore: $$ | q_ {n} (x) – f (x) | _ { infty} = sup_ {x in (a, b)} | q_ {n} (x) – f (x) | leq sup_ {x in (a, b)} | q_ {n} (x) – F (x) | leq sup_ {x in (a, b)} | p_ {n} (x) – F (x) | + | p_ {n} (0) | < frac { epsilon} {2} + frac { epsilon} {2} = epsilon $$

5) Therefore, we show that $ q_ {n} (x) to f (x) $ uniformly and in particular this means $ q_ {n} (0) = 0 $. So, according to the fundamental theorem of algebra, this implies $ q_ {n} (x) $ it's the way $ q_ {n} (x) = x ^ g (x) $, for some polynomials $ g (x) $.

**Thoughts:**

I am working hard to try to raise my mathematical thinking. As such, I have some questions about the process my teacher went through:

1) Where did the idea of defining this new function come from? $ F (x) $ comes from? He mentioned that he wanted to create an extension and I vaguely understand what it meant, but I don't see how he came to that conclusion. As in "we have our function $ f (x) $ and for Weirstrauss there is a sequence of approximate polynomials … "This is what went through my mind, but he obviously knew how to extrapolate even more from that.

3) Why define the polynomial? $ q_ {n} (x) $ in the way it was done? What was the impetus for this movement?

5) I see how the FTA can be applied at this stage, but I would never have thought to show the uniform convergence in this fashipn and then draw the conclusion that was drawn. It is as if he knew that this should be the process to arrive at the result before even solving the problem.

These things are tormenting my brain, because I didn't have a written solution beforehand and just noted that the question took about 15 minutes and solved it. I was working hard for a day and made no progress. What kind of questions should I ask myself between each step to improve my problem-solving / test writing skill set?