TITLE: Can a sequence of unbounded functions be uniformly convergent?
QUESTION [11 upvotes]: If I say have a sequence of functions $\displaystyle x^3 + 5\frac{x}{n}$.
I know that I can say it is point-wise convergent as $n$ goes to infinity to the limit function $x^3$ but can I simply say that as it isn't bounded it CAN NOT be uniformly convergent?

REPLY [8 votes]: Yes, you can have a sequence of unbounded functions that converges uniformly to an unbounded function.
Consider $f(x) = x^{2}$.  Given some $\epsilon > 0$, imagine drawing a bubble around $x^{2}$ of radius $\epsilon$.
Now let $f_{n}(x) = x^{2} + \frac{1}{n}$.  So $f_{n}$ is just $f$ shifted upward by $\frac{1}{n}$.
Clearly, $f_{n}$ converges to $f$ pointwise.  But look at the bubble around $x^{2}$ you drew.  Does there exist a point $N$ in the sequence $f_{n}$ such that for all points $n$ after $N$, the entire function $f_{n}$ is in the bubble around $f$?  The answer is yes, and hopefully you can see it if you drew a picture.  Because the answer is yes, $f_{n}$ converges uniformly to $f$, even though for all $n$, $f_{n}$ and $f$ are unbounded.
More generally, let $f(x)$ be any unbounded function and let $f_{n}(x)$ be defined as $f(x) - \frac{1}{n}$.  Then $f_{n}$ is also unbounded, and $f_{n}$ converges to $f$ pointwise (it is an excellent and easy exercise to prove this).