I know that, if a sequence has no upper bound, it doesn't necessarily mean that it is going off to infinity. However, I don't know how to prove that this isn't the case when my sequence is increasing.
I know that $u_n \le u_{n+1}$, and I want to say that $u$ has to be going to infinity, and therefore there's a point at which $u$ is positive, but I can't use that definition. Is there a more precise definition for a sequence with no upper bound to show that the title is true, or am I going completely the wrong way? Thanks.
Pick your favorite real number $L$.
$L$ is not an upper bound of the sequence, so there exists an index $N$ with $u_N > L$.
Since $\{u_n\}$ is increasing, $n \ge N \implies u_n \ge u_N \implies u_n > L$.
That is, for every $L \in \mathbb R$ there exists $N \in \mathbb N$ with the property that $$n \ge N \implies u_n > L.$$
This is the definition of $u_n \to \infty$.