Proof that fn converges uniformly

[kasperrepsak]

Homework Statement

\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 isn't an N for which if n > N the inequality in the definition of uniform convergence holds for all x.

Homework Equations

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

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

 

[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
<br /> M_n = \sup_{x \in I} |f(x) - f_n(x)|<br />
is such that M_n \to 0.

 

[kasperrepsak]

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