Prove that if a sequence is bounded, then limsup sn is a real number.

Expert Solution

Colby Messinger answered 2 years ago

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if...

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if and only if every subsequence of (an) has a convergent subsequence.

E.C. 2. (10 pts.) Suppose that (sn) is a sequence of real numbers such that sn...

E.C. 2. (10 pts.) Suppose that (sn) is a sequence of real numbers such that sn ≥ 0 for all n ∈ N. (a) Show that the set of subsequential limits of S satisfies S ⊆ [0,∞) ∪ {+∞}. (b) Is it possible for S = [0,∞) ? (Hint: apply Theorem 11.9.) Legible handwriting is a must

Sn = (1+(1/n))^n (a) Prove Sn is strictly increasing (b) bounded below by 2 and above...

Sn = (1+(1/n))^n (a) Prove Sn is strictly increasing (b) bounded below by 2 and above by 3 (c) Sn converges to e (d) Obtain an expression for e (e) Prove e is irrational

Let {an} be a bounded sequence. In this question, you will prove that there exists a...

Let {an} be a bounded sequence. In this question, you will prove that there exists a convergent subsequence. Define a crest of the sequence to be a term am that is greater than all subsequent terms. That is, am > an for all n > m (a) Suppose {an} has infinitely many crests. Prove that the crests form a convergent subsequence. (b) Suppose {an} has only finitely many crests. Let an1 be a term with no subsequent crests. Construct a...

Prove that if a sequence is bounded, then it must have a convergent subsequence.

Let A be a subset of all Real Numbers. Prove that A is closed and bounded...

Let A be a subset of all Real Numbers. Prove that A is closed and bounded (I.e. compact) if and only if every sequence of numbers from A has a subsequence that converges to a point in A. Given it is an if and only if I know we need to do a forward and backwards proof. For the backwards proof I was thinking of approaching it via contrapositive, but I am having a hard time writing the proof in...

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all...

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a. (b) Show that if sn ≤ b for all but finitely many n, then lim sn ≤ b. (c) Conclude that if all but finitely many sn belong to [a,b], then lim sn belongs to [a, b].

1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove...

1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove that if A is a bounded set, then A-bar is compact.

if (a_n) is a real cauchy sequence and b is also real, prove i) (|a_n|) is...

if (a_n) is a real cauchy sequence and b is also real, prove i) (|a_n|) is a cauchy sequence ii) (ba_n) is also a cauchy sequence

Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only...

Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only if every infinite subset of E has a point of accumulation that belongs to E. Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real numbers is closed and bounded if and only if every sequence of points chosen from the set has a subsequence that converges to a point that belongs to E. Must use Theorem 4.21 to prove Corollary 4.22 and there should...

Question