Definition talk:Pointwise Convergence

It doesn't actually matter whether $$N$$ is in $$\N$$ or $$\R$$. I deliberately put $$\R$$ when I initially posted this one because it makes any proofs relying on this definition technically easier - you don't have to ensure that the $$N$$ you are using has to be an integer. Would you be okay with me putting it back to $$\R$$ again (perhaps with a note of explanation)?

Quote from Keith Binmore: Mathematical Analysis: A Straightforward Approach (1977, Cambridge University Press): "Note: Some authors insist that $$N$$ be a natural number. This makes the definition of convergence a little more elegant but renders examples like the one above [a proof that $$1 + \frac 1 n \to 1$$ as $$n \to \infty$$] marginally more complicated. If we wanted $$N$$ to be a natural number in [the example given], we could not simply write $$N = \frac 1 \epsilon$$. Instead we should have to choose $$N$$ to be some natural number larger than $$\frac 1 \epsilon$$."

I'm with Binmore on this one. I don't even agree that insisting that $$N \in \N$$ does make the definition more elegant.