# Do I need induction to prove that this sequence is monotonic?

CoffeeNerd999
TL;DR Summary
If i'm proving ##y_n = \sup\{x_j | j \geq n\}## is decreasing, can this be done without invoking mathematical induction?
I think the initial assumptions would allow me to prove this without induction.

Suppose ##(x_n)## is a real sequence that is bounded above. Define $$y_n = \sup\{x_j | j \geq n\}.$$

Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n##
$$x_{j} \leq y_n.$$ So, ##y_n## is an upper bounded of ##(x_j)_{j=n+1}^\infty##. By definition, ##y_{n+1}## is the least upper bound of ##(x_j)_{j=n+1}^\infty##, so
$$y_{n+1} \leq y_n.$$ Since ##n## was chosen arbitrarily, this proves ##y_n## is monotone decreasing.

Am I correct that the using the supremum's definition makes using the induction principle suprifilous for this result?

Your proof is correct and your observations as well. You don't need mathematical induction here. The following is the general situation that you applied: $$\emptyset \neq A \subseteq B \implies \sup A \leq \sup B$$