(ii) Prove that if a sequence Xn converges to a limit I, then any subsequence of Xnalso converges to I. (7 marks)
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- Let fn be a bounded sequence of functions uniformly convergetn to f. Prove that f is bounded as well. Is the claim true if we replace the assumption of uniform convergence with pintwise convergence.Prove formally whether the given sequence has a limit, using the Cauchy criterium. If the sequence is properly divergent prove it formally too.Use Theorem 1 to determine the limit of the sequence or state that it diverges theorem 1: If lim x--> infinity f(x) exists, then the sequence an=f(n) converges to the same limit
- Show that the sequence converges and find its limit.Find the limit of the sequence (a,, i it is convergent. Otherwise, please type DIVERGENT(a) Let (an) be a bounded (not necessarily convergent)sequence, and assume lim bn = 0. Show that lim(anbn) = 0. Why arewe not allowed to use the Algebraic Limit Theorem to prove this?