[Math] Prove that a monotone sequence of real numbers is convergent if and only if it is bounded

Question

To prove for a decreasing sequence. I'm getting difficulties with the second part

1. First I assumed it is convergent and and prove that it is bounded.
2. Assume bounded and prove convergent
   I have used completeness property $\ X_n$ $\geq M$ for all $n \in N $ where M is the infimum

Answer 1

Hint: For any $\epsilon > 0$, $M+\epsilon$ is not a lower bound for the sequence. Hence there exists $X_n$ such that $$ M+\epsilon > X_n \geq M $$