For each n element N let fn 1 infinity rightarrow R be given

For each n element N let f_n: (1, infinity) rightarrow R be given by f_n (x) = 1/1 + x^n. Find the function f: (1, infinity) rightarrow R to which {f_n} converges point wise. Prove that the convergence is not uniform.

Solution

General method,

fix x =a

So (1/1+a^n) converges as n --> inf = 0

So it is a pointwise convergence

Compute |fn(x) - f(x)| = | 1/(1+x^n) | as f(x) converge to zero

The supremum of this over the domain = 1/2 not equal to zero

This is not zero which is not a uniform metric.So it diverges.

So we can say that convergence is not uniform