Show that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution How can I show (preferably using the Bolzano-Weierstrass theorem), that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution? I can see that it is true, but how can I show it?

I don't really understand the Bolzano-Weierstrass connection, but fixing $x_1 = 0$ (say) reduces the function to $x_2^2(1-x_2),$ which is obviously going to $-\infty$ as $x_2$ goes to $\infty.$
