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 enough, and consequently, \(\lim _{n \rightarrow \infty} f\left(x_{n}\right)=\eta\). If, on the other hand, \(\lim _{n \rightarrow \infty} f\left(x_{n}\right)=\eta\) whenever \(x_{n} \rightarrow \xi\), we must have \(\lim _{x \rightarrow \xi} f(x)=\eta\). Otherwise there would exist a positive \(\epsilon\) such that \(|f(x)-\eta| \geq \epsilon\) for some \(x\) arbitrarily close to \(\xi\); there would then also exist a sequence \(x_{n}\) converging to \(\xi\) for which \(\left|f\left(x_{n}\right)-\eta\right| \geq \epsilon\), but then \(\lim f\left(x_{n}\right)\) could not be \(\eta\).