Let an be a bounded sequence of numbers For each natural num

Let {a_n} be a bounded sequence of numbers. For each natural number n and each number x, define f_n(x) = a_0 + a_1x + a_2x^2/2! + ... + a_nx^n/n!. Prove that for each r > 0, the sequence of functions {f_n: [-r, r] rightarrow R} is uniformly convergent.

Solution

Let M be any bound for the sequence {a[n]:

               |a[n]|< M for all n.

Then |f_n(x)|< M e^r over the interval [-r,r].

So the sequence of functions converges by Weierstrass;s test