How to prove that the supremum of the subset of a nonempty and bounded above is equal to the function of the subset

Question

Let $$A \subset R$$ be non-empty and bounded above and let $$ B = \{ \sqrt2 a : a \in A \}.$$ Then, $$\sup B = \sqrt 2 \cdot \sup A. $$

Answer 1

[HINT] If $a\in A$ is such that $a-\varepsilon>x$ for every $x\in A$, then $ \sqrt{2}a-\sqrt{2}\varepsilon>\sqrt{2}x$ for every $x\in A$ and so $$ \sqrt{2}a-\sqrt{2}\varepsilon>y $$ for every $y\in B$.