Limit definition : the equivalence of Eduard Heine's with the (ε, δ)
I think Eduard Heine's definition of function limit is much intuitive than the prevailing (ε, δ) definition, while the latter is a bit convenient than the former in using. The following is a proof of the equivalence of the two, excerpted from Richard Courant, Fritz John, Introduction to Calculus and Analysis Volume I, Reprint of the 1989 edition, p82. The limit of a function can also be described completely in terms of limits of sequences. The statement \[ \begin{array}{l} \lim _{x \rightarrow \xi} f(x)=\eta \end{array} \] means that \[ \begin{array}{l} \lim _{n \rightarrow \infty} f\left(x_{n}\right)=\eta \end{array} \] for every sequence \(x_{n}\) with limit \(\xi\) (where it is assumed, of course, that the \(x_{n}\) belong to the domain of \(f\) ). For if \(\lim _{x \rightarrow \xi} f(x)=\eta\) and if \(\lim x_{n}=\xi\), then \(f(x)\) is arbitrarily close to \(\eta\) for \(x\) sufficiently close to \(\xi\); but \(x_{n}\) is sufficiently close to \(\xi\) if only \(n\) is large e...
