If $A$ and $B$ are nonempty, bounded subets of $\mathbb R$, then $ \boldsymbol{\sup (A\cap B) \leq \sup A}$.

Question

If the statement is true, then provide a proof. If the statement is false, then provide a counterexample.

From my understanding, I think this statement is false because the $\sup A$ has to be greater than the intersection of $\sup A \cap B$.

Answer 1

Hint

For $x\in A\cap B$, $$ x\leq\sup A $$ since $A\cap B\subset A.$

Answer 2

Hint: $A \cap B \subset A$. What does this tell you about the sup?