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.