Does a bounded monotonic sequence is convergent?

A bounded monotonic increasing sequence is convergent. We will prove that the sequence converges to its least upper bound (whose existence is guaranteed by the Completeness axiom). So let α be the least upper bound of the sequence.

Does bounded sequence converge?

No, there are many bounded sequences which are not convergent, for example take an enumeration of Q∩(0,1). But every bounded sequence contains a convergent subsequence.

Are all monotonic sequence bounded?

Only monotonic sequences can be bounded, because bounded sequences must be either increasing or decreasing, and monotonic sequences are sequences that are always increasing or always decreasing. The smallest value of an increasing monotonic sequence will be its first term, where n = 1 n=1 n=1.

Under what condition will a bounded sequence be convergent?

Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

Is a monotonic sequence always convergent?

If the sequence is both monotonic and bounded, then it is always convergent. 1. Show that the following sequences is monotonic. Is it an increasing or decreasing sequence?

What are the properties of convergent sequences?

Properties of convergent sequences. Bounded sequences. A sequence is bounded above if there is a number M such that an < M for all n. It is bounded below if there is a number m such that an > m for all n . If a sequence is bounded above and bellow it is called bounded sequence.

What is the difference between a bounded and a monotone sequence?

If a sequence (x n) converges it is bounded (you should proove it showing that every element except a finite number of them of the sequence is at distance at most 1 from the limit and then conclude). But on the other hand, if x n := (− 1) n n (n ≥ 1) then the sequence goes to 0 at infinity but it is not monotone.

How do you prove that a sequence converges?

If you want to prove the statement, if a sequence is monotone and bounded then it converges]