a/b, a/b+ c/d, a/b+ c/d+ e/f, etc.
say a/b+ c/d+ e/f+ g/h we have, g/h being between ±1, so is e/f+ g/h, so therefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h.
Now, if possible, let a/b+ c/d+ etc. be A/B at the limit; A and B being integers. Let
P = A c/d+ e/f+ etc., Q = P e/f+ g/h+ etc., R = Q g/h + i/k + etc.
P, Q, R, etc. being integer or fractional, as may be. It is easily shown that all must be integer: for
A/B = a/b+ P/A, or, P = aB - bA
P/A = c/d+ Q/P, or, Q = cA - dP
Q/P = e/f+ R/Q, or, R = eP - fQ
etc., etc. Now, since a, B, b, A, are integers, so also is P; and thence Q; and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1 and +1, it follows that the unlimited succession of integers P, Q, R, are each less in numerical value than the preceding. Now there can be no such unlimited succession of descending integers: consequently, it is impossible that a/b+ c/d+, etc. can have a commensurable limit.