Let x be a positive number, with $\displaystyle x^2>2.$
Let $\displaystyle y = x/2 + 1/x$
Show that $\displaystyle 0<y<x$, and that $\displaystyle y^2>2$. Explain why this shows that there is no smallest positive number whoes square is more than 2.
2. Just do the claims in turn. To show that $\displaystyle y < x$, show that $\displaystyle 1/x < x/2$. To show $\displaystyle y^2>2$, calculate $\displaystyle (x/2+1/x)^2$. I hope you can figure out the last part provided the rest is shown.