Uniformly continuous function (sequence)

[angst18]

Homework Statement

(This is my first post and I'm not sure why the Tex code isn't working, sorry).Suppose $f$is a positive continuous function on $[1,0]$.For each natural number$n$ define a new function$F_n$ s.t.
$$F_n(x) = \int_0^1 t^ne^{xn}f(t)dt$$

(a) Prove that $lim_{n\to\infty}F_n(x) = 0$ for all real $x$.
(b) Prove that the above limit is uniform on each bounded interval $[a,b]$.
(c) Determine with proof or counterexample wether or not the limit is uniform on $(-\infty, \infty)$.

Homework Equations

The Attempt at a Solution

So, I know that what I'm supposed to do for part (a) is to show that the limit is uniform so that I can bring it into the integrand and evaluate. I even know how to do this in when there's only one variable, but the addition of a 't' as well as an 'x' has me stymied.
I know I'm supposed to fix $f(t)$for $t\in [0,1]$ and $x$ (still not sure if i have to do the cases where $x$ is neg/pos) and then show that the limit is independent of $x$ and $t$, but there's something I'm not getting, or I'm doing it in the wrong order, because I'm basically totally stuck.
Thanks in advance for any help.

 

[SammyS]

Hello angst18. Welcome to PF.

To get Tex to work:

After selecting "Preview Post", you need to click on your browser's "Refresh" button. Otherwise the display will not show the updated Tex image. Apparently some memory location doesn't get updated unless you do this.