Exercise 2.8.3. (a) Prove that (tnn) converges. (b) Now, use the fact that (tnn) is a Cauchy sequence to argue that (snn
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- Exercise 2 8 3 A Prove That Tnn Converges B Now Use The Fact That Tnn Is A Cauchy Sequence To Argue That Snn 1 (39.65 KiB) Viewed 28 times
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Exercise 2.8.3. (a) Prove that (tnn) converges. (b) Now, use the fact that (tnn) is a Cauchy sequence to argue that (snn) converges. We can now set S = lim snn. n+00 In order to prove the theorem, we must show that the two iterated sums converge to this same limit. We will first show that S =ΣΣaij. i=1 j=1 Because {tmn: m, n E N} is bounded above, we can let B = sup{tmn: m, n E N}.
1X j=1 ΟΧΟ Theorem 2.8.1. Let {aij : i, je N} be a doubly indexed array of real numbers. If ΣΣ|aijl i=1 j=1 converges, then both ΣΙΣ, α;; and ΣΙΣ dij converge to the same value. Moreover, lim Sun =ΣΣaij =ΣΣΑ.). i=1 j=1 j=1 i=1 where Snn = ΣΕΙΣ=1 diy. = Proof. In the same way that we defined the rectangular partial sums smn above in equation (1), define tumn = ΣΣΙa;;). i=1 j=1 m η