I just got the definition of pointwise and uniform convergence for sequences of functions. I have not yet seen any theorems related to these notions, only the definitions. I have been playing around with examples to make sure I really understand them.
The first example I came up with was the sequence $f_n = x^{-n}$. First I chose these functions to be defined on [1,2] and it was easy enough to show that $f_n \to f(x) = 0$, pointwise. It was also pretty easy to see that on [1,2] the sequence would necessarily fail uniform convergence since no function in my sequence will ever budge at $x = 1$. But what if I chose the domain as (1,2)? I have been unable to prove that it is or is not uniformly convergent to $f(x) = 0$ on this domain. I conjecture that it will not be uniformly convergent since if we get close enough to $x = 1$, there should always be some $x$ that raised to a finite exponent will fail to be less than $\epsilon$, but I am having trouble constructing an argument.
To be honest I only tried for about 10 minutes, but I really should be working on other things right now. Thanks for any help, happy to provide more information or examples of what I have tried if needed.
Note that $f_n(\sqrt[n]{2}) = {1 \over 2}$. Hence $\sup_{x \in (1,2)} |f_n (x) - 0| \ge {1 \over 2}$.