1. (a) Let an→ 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let an → 4. Prove that (an) has a subsequence which converges either to 2 or -2. (c) Let a →→ 4. Prove from first principles (i.e. give a direct e-N proof) that |an| → 2.
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- 1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a, → 4. (b) Let a → 4. Prove that (an) has a subsequence which converges either to 2 or -2. (c) Let a → 4. Prove from first principles (i.e. give a direct e-N proof) that |an → 2.If 2n=1a, converges, and if a„ # 1 and a,n > 0 for all n, a. Show that E=1a, converges. b. Does E-1a,n/(1 – a„) converge? Explain.7. Let fn: R → R be a sequence of continuous functions which converges uniformly to a function f : R → R. Let (n) be a sequence of real numbers which converges to x E R. Show that fn(xn) → f(x).
- 2n 15. Let a= 3n +1 a. Determine whether (a,) is convergent. Answer Yes b. Determine whether a, is convergent. AnswerΣ (-1)** In+15 a. Absolutely Convergent b. Conditionally Convergent с. Divergent5. Prove or disprove. (a) Let 2 = (: € C :1 < |:| < 2). Then there is a sequence of polynomials that converges uniformly to f(s) = ! (b) Let Ja] < 1. If Then -.