(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that [tex]f_{n} = \frac{x}{\sqrt{1+nx^2}}[/tex] is uniformly convergent to 0 on all real numbers

2. Relevant equations

{f_n} is said to converge uniformly on E if there is a function f:E->R such that for every epsilon >0, there is an N where n>=N implies that | f_n(x) - f(x) | < epsilon, for all x in E.

3. The attempt at a solution

Let f(x) = lim n-> infty f_n(x), and let epsilon > 0. Then it is obvious, that if n>1, that as n -> infty, the limit goes to 0, and thus we would need to show that [tex]\frac{x}{\sqrt{1+nx^2}} < epsilon[/tex]| , which happens as long as n > [tex]\frac{\frac{x}{epslion}-1}{x^2}[/tex]. So, I feel like I got it, except for the 'obvious' statement that f(x) = 0. Am I doing this right? Thanks ahead of time.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Uniformly convergent on R

**Physics Forums | Science Articles, Homework Help, Discussion**