If it exists, determine the supremum of following set. Must prove answer.
If it exists, determine the supremum of following set. Must prove answer. S = (n/n +1:...
Let (an)nen be a bounded sequence in R. For all n e N define bn = sup{am, On+1, On+2,...}. (You do not have to show that the supremum exists.) (a) Prove that the sequence (bn)nen is a monotone sequence. (b) Prove that the sequence (bn)nen is convergent. (c) Prove or disprove: lim an = lim bre. 100 000
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact. (b) Prove that for any є > 0 there exists some N > 0 so that for any x E A we have (c) Prove that A is totally bounded. (d) Prove that A is compact (2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact....
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.
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...
1. Suppose that f : NR. If lim f(n+1) f(n) = L n-oo prove that lm0 S (n)/n exists and equals L 1. Suppose that f : NR. If lim f(n+1) f(n) = L n-oo prove that lm0 S (n)/n exists and equals L
Topology C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
(2) Define the set A by (a) Prove that for any N 20 the set is compact. (b) Prove that for any e>0 there exists some N 2 0 so that for any x A we have (c) Prove that A is totally bounded. d) Prove that A is compact.
18. Prove the infinite pigeonhole principle, that is, let S be an infinite set, n E Zt. Prove that no matter how the elements of S are partitioned into n parts, at least one of the parts must be infinite 18. Prove the infinite pigeonhole principle, that is, let S be an infinite set, n E Zt. Prove that no matter how the elements of S are partitioned into n parts, at least one of the parts must be infinite
The intersection detection problem for a set S of n line segments is to determine whether there exists a pair of segments in S that intersect. Give a plane sweep algorithm that solves the intersection detection problem in O(nlogn) time. Prove it only requires O(nlogn) time.
Can someone fully solve this for me please 5. (6 marks) For each of the following sets determine whether the supremum and infimum exist and if so, give the supremum and infimum. (You are not required to show any working for this question.) (a) Q (b) EN n+2 1,5 5. (6 marks) For each of the following sets determine whether the supremum and infimum exist and if so, give the supremum and infimum. (You are not required to show any...