Limit of a sequence of infinities

Question

I'm having trouble dealing with limits involving infinities. Suppose I have sequence $\{a_n\}_{n \in \mathbb{N}}$ and $a_n = \infty$ for all $n$. Then is the following true?

$$\lim_{n \to \infty} a_n = \infty$$

By definition, I would have to show that for all $\epsilon > 0$, there exists an $N$ such that $|a_n - \infty| < \epsilon$ for all $n > N$, but this is clearly not true since $\infty - \infty$ is not defined. If not, what's the limit then?

Answer 1

You seem to mix up between a couple of things. A sequence can have a limit. We say that $a_n$ converges to some limit $L$ and we write it $\lim_{n \to \infty} a_n = L$. If the sequence does not converge to a finite limit $L$ it can converge to infinity (some will say it diverges) and we write it like so: $\lim_{n \to \infty} a_n = \infty$. The formal definition for this kind of limit is as follow:

$\lim_{n \to \infty} a_n = \infty$ if for every $M\in R$ there exist $N$ such that for all $n>N \implies a_n>M$

However, elements of a sequence can not be $\infty$ because infinity is not a number, but rather a concept.