$\qquad\qquad\qquad\qquad\qquad X \preceq Y \preceq Z$ and $|X| = |Z|$. Prove that $|Y|=|Z|$.
Just started on cardinalities. Not sure about this one.
Am I right if I do something along the lines of: we define injective functions $f:X\to Y$ and $g:Y\to Z$. Composite $fg$ is bijective and, hence, surjective. It must be the case that $g$ is also surjective. By definition, we already know that $g$ is injective. Therefore, we have $g:Y\to Z$ is bijective $\implies |Y|=|Z|$.
Be a bit cautious, here. In general, knowing only that $|X|=|Z|$ and that $h:X\to Z$ is injective isn't enough to conclude that $h$ is bijective. (Consider the map $\Bbb N\to\Bbb N$ given by $n\mapsto n+1$, for example.) For finite sets, we can draw that conclusion, but if you haven't proved that yet, then you shouldn't use it as a result. Now, it's true that if $f:X\to Y$ and $g:Y\to Z$ are such that $g\circ f:X\to Z$ is surjective, then so is $g$. Hence, your conclusion is fine, so long as you know that an injection between finite sets of equal cardinality is a bijection.