Define the two sets:
$\Omega = \{(a,b) \in \mathbb{R}^2 \; | \; a>0, b>0\}, \; \; \; \; \overline{\Omega} = \{(a,b) \in \mathbb{R}^2 \; | \; a\ge0, b\ge0\}$
as well as the functions
$g: \overline{\Omega} \to \mathbb{R}, \; \; g(a,b) = \frac{a^p}{p}+\frac{b^q}{q} \; \; , \; a,b \ge 0\\ f: \overline{\Omega} \to \mathbb{R}, \; \; f(a,b) = ab, \; \; a,b \ge 0$.
Furthermore, let c > 0 and consider the sets
$M_c = \{(a,b) \in \Omega \; | \; g(a,b) - c = 0\}\;$ and $\;\overline{M_{c}} = \{(a,b) \in \overline{\Omega} \; | \; g(a,b) -c = 0\}$
The task is: show, that the set $\overline{M_{c}}$ is bounded.
I tried defining the two functions $g_1: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}, \; \; a \mapsto g(a,0) \text{ and } \\ g_2: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}, b \mapsto g(0,b)$
As $g_1, g_2$ are strictly monotonic increasing and unlimited, consequently the preimages $g_1^{-1}(c) = \{a_0\}$ and $g_2^{-1}(c) = \{b_0\}$ are bounded. However for this to work, I have to show, that for all $(a,b) \in \overline{M_c}$ the following inequality holds: $a \le a_0$ and $b \le b_0$. This would then imply the boundedness of the $\overline{M_c}$.
How could this be shown? And is my approach constructive?
Showing that $\overline M_c$ is bounded is trivial: If $(a,b)\in\overline M_c$ then$$\frac{a^p}p\le\frac{a^p}p+\frac{b^q}q= c,$$so $$a\le{(cp)}^{1/p}.$$Similarly $$b\le(cq)^{1/q}.$$So $\overline M_c$ is contained in a (bounded) rectangle.