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
So to prove it is uniform convergence on [0,1]:
lim [|sup(x^2 - (x-1/n)^2)|:x ele [0,1]] = lim[|sup(2x/n - 1/(n^2))|: x ele [0,1]] = lim[2n/(n^2) - 1/(n^2)] = 0. Thus uniform convergence by theorem or remark as they call it in the book.
So to prove it is pointwise convergence:
Do I just need to state lim as n-> inf (x-1/n)^2 = x^2 for all x ele [0,1]? Or is there anything more rigorous I need to do?
Thanks
yes. but the sup is outside of the absolute values
nope, that's it. just take the limit. if you want to make it rigorous, you can use the definition of a limit to prove that the limit is as you say, but that's completely unnecessary as far as i'm concerned--and i think your professor would feel the same way. if you're uncomfortable though, you can prove that the limit is in fact x^2 by the definition of the limitSo to prove it is pointwise convergence:
Do I just need to state lim as n-> inf (x-1/n)^2 = x^2 for all x ele [0,1]? Or is there anything more rigorous I need to do?
Thanks