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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!

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    Completely disagree with Tony about "sloppy" notation. It is perfectly fine. In fact, to me, from what is written, it is obvious that the limit exists. Your friend is just creating issues out of thin air. Of course, I still haven't been able to find references for this usage, but I am pretty sure I have seen it. One good thing about that notation is that it is language independent!2011-01-15

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If you are using the extended Reals, then the definition of convergence also includes the infinities.

If you are not using extended Reals, then the (usual) definition of convergence is restricted to finite reals, in which case the two sets you have are the same.

Even without extended reals, you could say a sequence converges to infinity, but that is not common I believe (of course, I might be wrong about that). In which case, the sets could be different.

So, the real issue is: What definition of convergence are you using?

IMO, you are right. Just writing $\lim... \lt \infty$ carries the implicit assumption that it exists. I believe many authors use this as a convenient way to say the limit exists, in fact (I will try to update with references when I find them).

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    Thanks in advance, I'd appreciate it if you post a reference. I tried to find the answer on the internet but to no avail.2011-01-16