Proving a given sequence is a delta sequence ~

[retrofit81]

Hi! I'm in a mathematical ecology class and we're working with delta sequences.

I'm trying to show that

delta_n(x) = n if |x| <= 1/2n
= 0 if |x| > 1/2n

is a delta sequence.

----Definition of a delta sequence---------------------------------------
Suppose delta_n is a sequence of functions with the property that

lim (int delta_n*f(x) dx, -infinity, infinity) = f(0)
n->inf

for all smooth, absolutely integrable functions f(x). Then delta_n is a delta sequence.
-----------------------------------------------------------------------

I thought I could start it by inserting the function into the definition, breaking up the resulting integral, and taking some limits (after some integration by parts, possibly) -- but that's gotten me nowhere. Perhaps I'm missing something there?

A "hint" that comes with the problem is:
Apply the Mean Value Theorem to a function of the form
F(x) = int( f(t) dt, a, x).

I'm not even sure how this hint applies.

I've been racking my brain a couple days and would totally appreciate some guidance! :)

Respectfully,

Michael

 

[Galileo]

Since f is a nice function, let F be an antiderivative of f. Then you can just evaluate the integral in terms of F (and n). Now you need to evaluate the limit and end up with f(0) somehow.