I've been trying, with no luck so far, to get my head around the differnce between pointwise and uniform convergence.

The definition which I am using for Uniform convergence is:

We say converges to uniformly on a set if

Where are some constants such that as

I understand this definition, but what I am having trouble with is seeing how any convergence sequence of function could fail to satisfy this!

Assume but not uniformly.

Now we fix and calculate for each possible choice of . There must be a value for which gives the greatest value (i.e. for which is furthest away from . So take this value and denote is for each choice . Then for each we have certainly satisfied .

Moreover this sequence must converge to 0 as have by assumption that . So converges to uniformly on a set . Contradicting our assumption that it converged but not uniformly!

Clearly there must be something very wrong with the above argument but I can't see what it is.

thanks for any help or explanation!