Examples of Sequences

Question

Can someone give examples for these types of sequences?

1. A sequence that is monotone but not convergent
2. A sequence that is not bounded but is convergent
3. A sequence that is monotone but not Cauchy
4. A sequence that is monotone and bounded, but not Cauchy

For 1. I think one exists, but for 2. and 4., not one exists; for 3. I think one might exists as well

I am only dealing with $\mathbb{R}^2$ here though

Answer 1

1) The sequence is necessarily unbounded. For example the sequence $(n)$

2) Every convergent sequence is bounded.

3) Find an unbounded sequence in $\mathbb R$. For example the sequence $(n)$.

4)The space must be incomplete.

Answer 2

I'll answer here for sequences in $\Bbb{R}$ (or subsets thereof), since it is ordered and hence monotonicity makes sense.

1) Consider the sequence $1,2,3,\ldots$

2) This doesn't exist. Only a finite number are a distance greater than $\epsilon$ from the point it converges to. A finite number of finite real numbers have a maximum, which serves as a bound.

3) The example from 1) works here as well.

4) All monotone bounded sequences are Cauchy. To see this let $x_n$ be a sequence assume that for every $\epsilon > 0$ and every $N > 0$ there exists $n,m > N$ such that $|x_n - x_m| > \epsilon$. You can then construct a subsequence $x_{n_i}$ containing pairs $x_{n_i}, x_{n_{i+1}}$ which each are a distance greater than a fixed $\epsilon$ from one another, which contradicts boundedness.