EXERCISE 2.8.7 (Cauchy Products)

I added the theorem at the top for reference

Theorem 2.8.1. Let {aij : i.] EN) be a doubly indexed array of real numbers If converges, then both i-143 1 aij and -1 aij converge to the same value. Moreover, oo o0 O 00 lim snn- aij i-l j-1 -14-1 aij where s,in be that the sum may Exercise 2.8.7. Assume that rg absolutely to A, and converges absolutely to B. (a) Show that the iterated sum pol 001 laibl converges so that we may apply Theorem 2.8.1. (b) Let snn- y-1ab, and prove that lim Sn AB. Conclude that k-2 j-1 i 1 i=1 j=1 where, as before, de aib-1 + abi-2t+a-1b