What exactly does Heine do?

[twoflower]

Hi all,

when our teacher shows us the computing of some limit of sequence, he does this:


\lim_{n \rightarrow \infty} \frac{n + n - n + 2*n}{\sqrt{n + 1}} =^{Heine} \lim_{x \rightarrow \infty} \frac{x + x - x + 2*x}{\sqrt{x + 1}}


He just switches the variable letters from 'n' to 'x' and claim the limit to be limit of the function. I don't understand the idea..We had Heine's theorem at the very beginning of limit of functions, it has something to do with the relationship between sequences and functions, but I THINK it doesn't (at least explicitly) say us to switch letters :)

Thank you for the explanation.

[HallsofIvy]

"Switching Letters" is purely cosmetic. You COULD use x to represent only integer values or use n to represent a real variable. It is, however, "traditional" (and so more familiar) to use n to represent integer values, as in a sequence, and use x to represent real variables, as in a function defined on R.

What you call "Heine" is just stating that limx->cf(x)= L then approaching c by any sequence of numbers (i.e. limn->inf(xn)) must also have L as a limit. In particular, if limx->inff(x)= L, then the sequence taking x to be only integer valued must also converge to L.