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 27 times

Both parts of 2.8.3, please. Use (tnn) as defined in thm 2.8.1.
Thank you!
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