True Here is the proof If s n A and s n t n B thent n s n t n s n B A 4 Not

True here is the proof if s n a and s n t n b thent n

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3. True. Here is the proof: IfsnAandsn+tnB, thentn= (sn+tn)snBA.4. Not true. A counterexample:sn= 0, andsntn= 0·n= 0 converges, buttn=ndiverges.4.2-9* Prove Theorem 4.2.12: Suppose that(sn)and(tn)are sequences suchthatsntnfor allnN.1. Iflimsn= +, thentn= +.2. Iflimtn=−∞, thenlimsn=−∞.Proof: 4.2-12* Suppose that(sn)converges tos.Prove that(s2n)converges tos2directly without using the product formula of Theorem 4.2.1(c). 18
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Since ( s n ) converges, it is bounded: | s n | M for all n ; and for the , there is N > 0 such that | s n s | < | s | + M , n > M. Then | s 2 n s 2 | = | s + s n || s s n | ( | s | + | s n | ) | s n s | ( | s | + M ) | s n s | < ( | s | + M ) · | s | + M = , n > N. 4.2-14* Prove lim 1 n 1 n +1 = 0 . Proof: To show: For any > 0, there exists N > 0 such that 1 n 1 n + 1 < , n > N. Consider | 1 n 1 n +1 0 | = 1 n 2 + n < 1 n 2 < 1 n . For the , we take N > 1 so that the desired inequality holds for any n > N .
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  • Natural number, lim sn

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