Let (an) be the sequence given byan = 1√2n + 1 − 1√2nProve from first principles that (an) is Cauchy.

Let (an) be the sequence given byan = 1√2n + 1 − 1√2nProve from first principles that (an) is Cauchy. [5 marks](b) Suppose that (bn) is a sequence of integers which is Cauchy. Prove that(bn) is eventually constant, i.e. bn = bm for all sufficiently large m and

. Prove, using the ϵ − N definition oflimit, that the sequence (an) given byan = n2n − 1converges to 12 .2. Prove using the definition of limit thatthe sequence (an) given byan = 1(3n − 1)converges to 0.3. Evaluate the following limits(a) limn→∞n4 + 3n1 + n3 + n5 .(b) limn→∞3n − 13n − 3n−1(c) limn→∞√n + 1 − √n(d) limn→∞ nr4 − 1n − 2!4. Let (an) be a sequence. Show that iflimn→∞ an exists, then (an) is bounded.5. Show that the sequence (an) given byan = n2n + 1does not converge.6. Use the sandwich theorem to findlimn→∞1(3n − 1)7. (a) Let n ∈ N. Define functions f, gbyf (x) = (1 + x)nandg(x) = 1 + nxShow by induction that for all nat-ural numbers n ≥ 1 and real num-bers x ≥ −1f (x) ≥ g(x).(b) Sketch graphs f and g case for thecase n = 3.(c) Let a > 1. Show that the sequence(an) is unbounded above.(d) If |a| < 1, show that limn→∞ an = 08. Let x ∈ R. Define a sequence of partialsums by (sn) bys1 = 1s2 = 1 + xs3 = 1 + x + x2... = ...sn = 1 + x + x2 · · · + xn−1Show that (sn) converges if and only if|x| < 1. Find the limit of the sequence(sn) if x = 910 .9. Evaluate:(a) limn→∞1√n(b) limn→∞5n + (−1)n4n(c) limn→∞n34n(d) limn→∞√n√n − 2(e) limn→∞sin n + n3n3(f) limn→∞(−1)n√n1https://pennance.us1

A sequence (sn) is defined by s0 = s1 = 1 and sn+2 = √sn+1 + sn for all n ≥ 0.(a) Using induction, show that 0 < sn < 2 for all n. [2 marks](b) For n ≥ 4, show thats2n − s2n−1 = (sn−1 − sn−2) + (sn−2 − sn−3).Hence show that (sn) is increasing. [3 marks](c) Explain why (sn) is convergent, and calculate its limit

If (xn) and (yn) are two Cauchy sequences, prove from first principles that(xn + yn) is Cauchy. [6 marks](b) Show that (an) is Cauchy, where an = 2n/3n. Clearly state any resultsused from lectures or the Supplementary Materials

Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n.