(b) Theorem: Assume {a} is a sequence which converges to a and (b) is a sequence which converges to b, then the sequence

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B Theorem Assume A Is A Sequence Which Converges To A And B Is A Sequence Which Converges To B Then The Sequence 1 (25.18 KiB) Viewed 37 times

(b) Theorem: Assume {a} is a sequence which converges to a and (b) is a sequence which converges to b, then the sequence {an + b} converges to a + b. Proof: Let > 0 be given. (1) Then there exists N₁ EN such that lan -a <for all n > N₁. (2) (3) and there exists N₂ € N such that bn - b < for all n > N₂. Let N= max(N₁, N₂}. Thus an + bn(a+b)| <e for all n ≥ N. (5) (i) Explain why (2) and (3) are true. (ii) Explain why (5) is true. (iii) Why does (5) prove the theorem? [12 marks]