show sets are or aren't bounded in an linear ordered set

Question

Let $X$ be a linearly ordered set, and $A\subset Y\subset X$.

• Can it happen that $A$ is bounded in $X$, but not in $Y$?

• Can it happen that $A$ is bounded in $Y$, but not in $X$?

I'm thinking that they both cannot happen but I'm not sure of an explanation

Answer 1

• We can take $A:=\mathbb{N}$, $Y:=[0,\infty)$ and $X:=[0,\infty]$, where $\forall x\in Y, x<\infty$. Then $A$ is bounded in $X$ (by $\infty$) while $X$ is not bounded in $Y$.

• If $y_0\in Y$ is a bound for $A$ in $Y$. Then $y_0\in X$. Therefore $A$ is bounded by $y_0$ in $X$.