Comparing supremum and infimum of two sets

Question

Let X,Y⊆ℝ be two non-empty sets. Prove that if ∀x∈X ∀y∈Y x≤y, then supX and infY exist and supX≤infY.

in this question, I said pick y'∈Y and since x≤y' ∀x∈X, then X is bounded above and by completeness axiom there exists supX. And for infY i did it similarly but i dont know whether my proof is true. And i dont know how to prove second part of the proof.

Answer 1

HINT: $$ \forall x\in X\forall y\in Y\, x\leq y\implies \forall y\in Y \sup X\leq y $$