I'm having an argument about what the notation of $\lim$ means.
Assume you have $f_n: X \rightarrow \mathbb{R}$. Are the following two sets equal:
\{ x \ |\ (f_n(x)) \ \text{converges} \} = \{ x \ |\ -\infty \lt \lim_{n \rightarrow \infty} f_n(x) \lt \infty\}
Edit:
You can assume the reals are not extended and that convergence means that the limit exists and is finite.
The person I'm arguing with insists that what I wrote on the right makes no sense because the limit might no be defined. My argument is that undefined in this case is ruled out by the notation, since you cannot write "undefined < something" so the "<" implies that the limit is defined and therefore that the two sets are equal.
Thanks for clarification!