How can I mathematically prove this?

Question

How can I mathematically prove that P1 and P2 will have the same value of $\eta$ at optimality? Although it seems clear from the intuition. I am looking for proof in the language of mathematics, not in English.

P1 \begin{align} \max_{\eta,x_1,x_2} \eta\\ \eta\leq \dfrac{x_1}{x_2}\\ \eta \leq x_2\\ 0\leq x_1\leq A\\ 0\leq x_2\leq A \end{align} P2 \begin{align} \max_{\eta,x_2} \eta\\ \eta\leq \dfrac{A}{x_2}\\ \eta \leq x_2\\ 0\leq x_2\leq A \end{align}

Answer 1

P1. We have $\eta^2\le \frac {x_1}{x_2}x_2=x_1\le A$, so $\eta\le \sqrt{A}$ and the equality is achieved when $\eta=x_2=\sqrt{x_1}=\sqrt{A}$.

P2. We have $\eta^2\le \frac A{x_2}x_2=A$, so $\eta\le \sqrt{A}$ and the equality is achieved when $\eta=x_2=\sqrt{A}$.