Set theory: partially ordered set convex set question A subset $C$ of a partially ordered set is called convex if it satisfies the following conditions: if $a \in C, b \in C$ and $a \le x \le b$, then $x \in C$. Let $f:A \to B$ be an increasing function, where $A$ and $B$ are partially ordered sets. Prove that if $C$ is a convex subset of $B$, then $f^{-1}(C)$ is a convex subset of $A$.
From What I know, I understand that 
Since $C$ is a convex subset of $B$ and $f$ is an increasing function,
$\forall x,y \in A, x \le y \implies f(x) \le f(y)$
We need to show that $f^{-1}(C)$ is a convex set. But does it mean I have to show there is a $y \in f^{-1}(C)$?
I am lost with this. Any help would be appreiciated.

Let $a, b \in f^{-1}(C)$, and $x \in A$ be such that $a \leq x \leq b$.
Then $f(a) \leq f(x) \leq f(b)$.
But $f(a), f(b) \in C$ and thus, $f(x) \in C$, yielding $x \in f^{-1}(C)$.
