Originally Posted by

**tbyou87** Hi,

Let fn(x)=(x-(1/n))^2 for x ele [0,1].

1) Show that the sequence of functions {fn} converges to some function pointwise on [0,1] and write the limit function.

2) Does {fn} converge to f uniformly on [0,1]? Prove your idea.

So, f(x) = x^2 pointwise on [0,1].

Then I wrote a proof but i'm not sure if it is for proving it is pointwise or uniform convergence:

Let eps>0. Choose N = 1/(sqrt(eps)). Then n> N => n > 1/(sqrt(eps)) => 1/(n^2)< eps => |-2x/n +1/(n^2)|< eps => |(x-1/n)^2-x^2|<eps.

Now i'm wondering how to do the other one. I don't really see also the difference between both definitions:

Pointwise:

for each esp > 0 and x ele S there exists N such that |fn(x)-f(x)|< eps for n>N.

Uniform Convergence:

for each eps > 0 there exists a number N such that |fn(x)-f(x)|<eps for all x ele S and all n > N.

Thanks