Define A:= { 5 + [(n-2)/n] , n€N } find sup(A) and inf(A).
2. If S:= {1/n - 1/min, me N}, find inf S and sup S. 4. Let S be a nonempty bounded set in R. (a) Let a > 0, and let aS := {as : S ES). Prove that inf(as) = a infs, sup(as) = a sup S. (b) Let b <0 and let b = {bs : S € S}. Prove that inf(bs) = b supS, sup(bs) = b inf S. 6. Let X be a nonempty set and...
5a) (5 pts) Find lim inf (xn) and lim sup (rn), for rn = 4 + (-1)" (1 - 2). Justify your answer 5b) (5 pts) Find a sequence r, with lim sup (xn) = 3 and lim inf (x,) = -2. 5c) (10 pts) Let {x,} be a bounded sequence of real numbers with lim inf (x,) = x and lim sup (x,) = y where , yER. Show that {xn} has subsequences {an} and {bn}, such that an...
4. (5 points) For the following sequences, determine lim inf an and lim sup an: Justify your reasoning: (a) (2 points) an = cos (), n E N. (b) (3 points) an = 2 + n+1(-1)", n E N.
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an (also denoted lim an) to be --+ n+ l.u.b. {z ER: an > & for an infinite number of integers n} and define lim inf an (also denoted lim an) to be g.l.b. {ER: An <for an infinite number of integers n}. Prove that lim inf an Slim sup an, with the equality holding if and only if the sequence converges. 19. Let ai,...
3. (a) Given n e N, prove that sup{.22 : 0<x<1} = 1 and inf{.22n: 0<x<1} = 0. (b) Find the supremum of the set S = {Sn: ,ne N}. Give a proof.
Problem 4 [20] Suppose A C R is a non-empty set, and sup(A), inf (A) exist. Show that sup(A) inf (A) if and only if |A. Problem 5 [25]
n+1 2. Let an = (-1)"-for n > 1. (a) What is sup(fas, ab,...])? (b) What is supa,)? (d) What is liminfan-sup ((inf((an.an+1, )) | n = 1,2.. ..))?
if A is a bounded set, then sup(A),inf(A) is in the
closure of A. can you prove this using Hein Borel theorem?
HiW If Ais a haudel et Can then supA), inA) E A as a that i fornr
16. lim n-> inf
24. Find the limit of the sequence
25. lim n-> inf
. limit=
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Separate each answer?
5) Define the supremum of a bounded above set SCR. 6) Define the infimum of a bounded below set SCR. 7) Give the completeness property of R 8) Give the Archimedean property of R. 9) Define a density set of R. 10) Define the convergence of a sequence of R and its limit. 11) State the Squeeze theorem for the convergent sequence. 12) Give the definition of increasing sequence, decreasing sequence, monotone se- quence. 13) Give the...