A simple proof of Euler's continued fraction of e^{1/M}

A continued fraction is an expression of the form f0+ g0 f1+g1 f2+g2 and we will denote it by the notation [f0, (g0, f1), (g1, f2), (g2, f3), … ]. If the numerators gi are all equal to 1 then we will use a shorter notation [f0, f1, f2, f3, … ]. A simple continued fraction is a continued fraction with all the gi coefficients equal to 1 and with all the fi coefficients positive integers except perhaps f0. The finite continued fraction [f0, (g0, f1), (g1, f2),…, (gk –1, fk )] is called the kth convergent of the infinite continued fraction [f0, (g0, f1), (g1, f2),…]. We define [f0, (g0, f1), (g1, f2), (g2,f3),...] = lim [f0, (g0, f1), (g1, f2),..., (gk-1,fk)] if this limit exists and in this case we say that the infinite continued fraction converges.