# How do you prove that a Cauchy sequence is bounded?

## How do you prove that a Cauchy sequence is bounded?

Proof 1

1. Let ⟨an⟩ be a Cauchy sequence in R.
2. Then there exists N∈N such that:
3. for all m,n≥N.
4. In particular, by the Triangle Inequality, for all m≥N:
5. So ⟨an⟩ is bounded, as claimed.
6. Let ⟨an⟩ be a Cauchy sequence in R.
7. Then there exists N∈N such that:
8. for all m,n≥N.

### Is every Cauchy sequence is convergent?

Theorem. Every real Cauchy sequence is convergent.

Can unbounded sequence be Cauchy?

If a sequence is a Cauchy sequence, it has to be bounded and it cannot have an unbounded subsequence.

Is every Cauchy sequence closed?

A set F is closed if and only if the limit of every Cauchy sequence (or convergent sequence) contained in F is also an element of F. Proof. Now assume that the limit of every Cauchy sequence (or convergent sequence) contained in F is also an element of F. We show F is closed.

## Is every bounded sequence monotonic?

Only monotonic sequences can technically be called “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.

### Does bounded imply Cauchy?

If a sequence (an) is Cauchy, then it is bounded. If a subsequence of a Cauchy sequence converges to x, then the sequence itself converges to x.

Is a sequence bounded?

A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K’, greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K’.

Are Cauchy sequences continuous?

It maps Cauchy sequences to Cauchy sequences but it’s not uniformly continuous on the whole real line. Take for X the disjoint union of a sequence An of two point sets.

## Why every convergent sequence is bounded?

Every convergent sequence of members of any metric space is bounded (and in a metric space, the distance between every pair of points is a real number, not something like ∞). If an object called 11−1 is a member of a sequence, then it is not a sequence of real numbers.

### Is every sequence bounded?

Proof: Every sequence in a closed and bounded subset is bounded, so it has a convergent subsequence, which converges to a point in the set because the set is closed.

Is every bounded sequence converges?

Every bounded sequence is NOT necessarily convergent.

Is every monotone sequence Cauchy?

If a sequence (an) is Cauchy, then it is bounded. Our proof of Step 2 will rely on the following result: Theorem (Monotone Subsequence Theorem). Every sequence has a monotone subsequence. If a subsequence of a Cauchy sequence converges to x, then the sequence itself converges to x.