Let x be an accumulation point of {an|n ∈ N}. Prove that there exists a subsequence of {an} ∞ n=1 that converges to x.

Q: Let (Sn) be an unbounded sequence. Prove that there exists a subsequence Sm with the property that…

A: To determine: Sm with the property that Sm converges to infinity.

Q: (d) For each N > 0, there is ɛ > 0 such that an - a| N +ɛ.

A: Solution:Since the subsequence {an} converges to a ∈ X, we know that for all ε > 0, there exists…

Q: Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is…

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Q: 00 Show that converges if [r| < 1. п: п

A: The given series is

Q: 2. Construct a sequence of unbounded functions In : [0,1] → R that converges pointwise to a bounded…

A: Step 1:

Q: For each nЄ Z+ let fn : R → R be the function пх fn(2) n|x| +1° (a) Prove that (fn) converges…

A: Given the sequence of functions  defined by .(a)Consider the limit,This is true for every 0≠x∈R.…

Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

A: Given that,  The functions fn and f are integrable on a,b. fn converges to f on uniformly a,b. an…

Q: a. Show that if two power series E-oa„x" and E-o b,x" are convergent and equal for all values of x…

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Q: Assume you have given a sequence c,, with non-zero values (cn #0 for n = 1,2,...) that fulfils the…

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Q: b) Suppose (xn) does not converge to 0. Then there exists e > 0 such that all except for finitely…

A: Let us suppose the following sequence xn=1,0,1,0,1,0,..... We have to see that the odd terms are 1…

Q: (a) Prove that (2x + 2) > 1 for all x > 0. (b) Let so > 0, and sn+1= } (2s, + ) for all n > 0. Prove…

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Q: Show that X is complete if and only if every absolutely convergent series in X converges.

A: To Show : X is complete if and only if every absolute convergent series in X converges.

Q: Let fn be a bounded sequence of functions uniformly convergetn to f. Prove that f is bounded as…

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Q: For eksample Xn = n² (n+1)! investigate whether the serles Σ 8 n=1 Xn convergent Or divergent. Prove…

A: We will use ratio test to check whether given series is convergent or not.

Q: Suppose (an) is a sequence of positive real numbers such that an+1 Prove that liman (an) even…

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Q: (3) Let {an} be a sequence in R that has a subsequence {an;} satisfying |ani — ani+1| ≥c for some…

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Q: Prove that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is…

A: Given function fn:ℝ→ℝ sequence of continuous function. fn is uniformly convergent on ℚ To prove: fn…

Q: Prove that every conditionally convergent series of real numbers that does not converge absolutely…

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