Question

Prove that a non-decreasing (resp. non-increasing) sequence which is not

bounded above (resp. bounded below) diverges to +∞ (resp. to −∞).

Answer 1

Solution:

To prove: A non-increasing sequence which is not bounded below diverges to −infinity.

Proof: Let $\left\{a_{n}\right\}$ be a monotonically non-increasing sequence.

$\Rightarrow \quad a_{n+1}<a_{n} \forall n \in {N}$

$\left\{a_{n}\right\}$ is not bounded below.

$\Rightarrow \quad$ For any $M \in {R}, \exists m \in {N}$ such that $a_{k}<M \forall k>m$ .

$\Rightarrow \quad$ $\lim _{n \rightarrow \infty} a_{n}=-\infty$

$\Rightarrow \quad$ A monotonically decreasing sequence that is not bounded below diverges to negative infinity.

Similarly, we can do other part.

To prove: A non-decreasing sequence which is not bounded above diverges to +infinity.

Proof: Let $\left\{a_{n}\right\}$  be a monotonically non-decreasing sequence.

$\Rightarrow \quad a_{n+1}>a_{n} \forall n \in {N}$

$\left\{a_{n}\right\}$ is not bounded above.

$\Rightarrow \quad$ For any $M \in {R}, \exists m \in {N}$ such that $a_{k}>M \forall k>m$

$\Rightarrow \quad \lim _{n \rightarrow \infty} a_{n}=+\infty$

$\Rightarrow \quad$ A monotonically increasing sequence that is not bounded above diverges to positive infinity.