Join of monotonic functions is monotonic

Question

Given two monotonic functions $F$ and $G$ on a complete lattice, how would I go about proving that the function:

$$ H : x \mapsto F(x) \vee G(x) $$

is monotonic (i.e. $a \leq b \implies H(a) \leq H(b)$)?

Answer 1

Let $a \le b$. Then $F(a)\le F(b) \le F(b) \vee G(b)$ and $G(a) \le G(b)\le F(b) \vee G(b)$. Thus $F(b) \vee G(b)$ is an upper bound of $\{F(a),G(a)\}$. Since $F(a) \vee G(a)$ is the least upper bound of $\{F(a),G(a)\}$ you have $F(a) \vee G(a) \le F(b) \vee G(b)$.