bounded

Def for R

For sequence $(x_n)$, we can say it’s bounded if

$$(\exists M>0)(\forall n\in \mathbb{N})|x_n|<M$$

Def 23.5

A set $S\in X$ is called bounded if there

$$(\exists R>0)(\exists x\in X)S\subset B_R(x)$$

Where $B_R$ is a Delta-ball.

Def

If we say a function is bounded, we can say

$$(\exists M>0)(\forall x)|f(x)|<M$$