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 $\displaystyle f_{m} $ converges to $\displaystyle f$ uniformly on a set $\displaystyle A$ if

$\displaystyle | f(z) - f_{m}(z)| \leq C_{m},\forall z \in A\\ $

Where $\displaystyle C_{m}$ are some constants such that $\displaystyle C_{m} \to 0$ as $\displaystyle m \to \infty $

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 $\displaystyle f_{m}(z) \to f(z) $ but not uniformly.

Now we fix $\displaystyle m$ and calculate $\displaystyle | f(z) - f_{m}(z)| $ for each possible choice of $\displaystyle z \in A$. There must be a value for $\displaystyle z$ which gives the greatest value (i.e. for which $\displaystyle f_{m}(z)$is furthest away from $\displaystyle f(z)$. So take this value and denote is $\displaystyle C_m$ for each choice $\displaystyle m$. Then for each $\displaystyle m$ we have certainly satisfied $\displaystyle | f(z) - f_{m}(z)| \leq C_{m}, \forall z \in A\\ $.

Moreover this sequence $\displaystyle C_{m} $ must converge to 0 as have by assumption that $\displaystyle f_{m}(z) \to f(z) $. So $\displaystyle f_{m} $ converges to $\displaystyle f$ uniformly on a set $\displaystyle A$. 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!