# Proof that fn converges uniformly

1. Dec 16, 2012

### kasperrepsak

1. The problem statement, all variables and given/known data

$\frac{x}{1+n^2*x^2}$ I must show if this converges uniformly or that it doesnt. So i must show that there is an N or that there isnt an N for which if n > N the inequality in the definition of uniform convergence holds for all x.

2. Relevant equations

http://en.wikipedia.org/wiki/Uniform_convergence

3. The attempt at a solution
Using point convergence one can easily see that the function converges to 0 for each x. So this can be used to see if there is an N for which |$\frac{x}{1+n^2*x^2}$|<$\epsilon$

2. Dec 16, 2012

### pasmith

It may be best to use the equivalent definition (given on the wikipedia page) that $f_n(x) \to f(x)$ uniformly on $I$ if and only if
$$M_n = \sup_{x \in I} |f(x) - f_n(x)|$$
is such that $M_n \to 0$.

3. Dec 16, 2012

### kasperrepsak

thank you, i solved it using that definition and by finding the maxima of fn by differentiation.