What is infinity added to itself a countably infinite number of times? Intuitively, it seems to me that $$\sum_{n=1}^\infty \infty = \infty \cdot \infty = \infty,$$
because $$ \sum_{k=1}^n \infty = n \cdot \infty = \infty,$$
but I don't know how to prove it. I asked my professor, and he said I could just use the fact that $\infty + \infty = \infty$, and I see that I can extend this by induction to the case of a finite sum, but I don't see how I can use it for the countably infinite sum.
Some context: I am trying to solve a homework exercise in my measure theory course, and I need to show that a given set function is an outer measure, but I am stuck on this step. (I also don't understand why $\infty \cdot \infty = \infty$, so it'd be great if you could explain that too, but it's not my main question.)
You're working in the extended real numbers, which are constructed by taking $\mathbb{R}$, sticking in two more elements $\infty$ and $-\infty$, and writing down a bunch of definitions stating how to do arithmetic with them. (In particular, $\infty \cdot \infty = \infty$ is a definition of how the operator $\cdot$ extends to using $\infty$ as an argument.)
The extended reals have a natural topology, whose exact definition is not really important here. But for instance, you would want it to be defined in such a way that the sequence $1,2,3,\dots$ is convergent and its limit is $\infty$.
Anyway, whenever you have a space on which addition makes sense and that has a topology, a natural way to define an infinite sum $\sum_{n=1}^\infty a_n$ is as the limit of the sequence partial sums $S_k = \sum_{n=1}^k a_n$, provided the limit exists. In this case, when $a_n = \infty$ for every $n$, we have $S_k = \sum_{n=1}^k \infty = \infty$ for every $k$. So $\{S_k\}$ is a constant sequence; in any reasonable topology its limit should be $\infty$. So that is how we interpret the infinite sum $\sum_{n=1}^\infty \infty = \infty$.