Question

All strictly monotonically decreasing sequences are convergent.

True or false

Answer 1

False. A sequence $\{a_n\}_{n\in \mathbb{N}}$ is called strictly monotonically decreasing if and only if $a_{n+1} < a_n$ for all $n\in \mathbb{N}$. Since $-(n+1) = -n-1 < -n$, the sequence $\{-n\}_{n\in \mathbb{N}}$ satisfies this definition and is strictly monotonically decreasing. Assume that it converges to some $a\in \mathbb{R}$. Then there is $n_0\in \mathbb{N}$ such that $a-1 < -m < a+1$ for all $m\geq n_0$. By the Archimedean property of the real numbers, there is $n_1\in \mathbb{N}$ such that $-n_1 \leq a-1$. Letting $n_2 = \operatorname{max}(n_0, n_1)$, we have that $a-1 < -n_2$ and $-n_2 \leq a-1$, a contradiction. Therefore, $\{-n\}_{n\in \mathbb{N}}$ is not convergent.