<s>[INST] Between which two consecutive integer numbers is $\sqrt{2}+\sqrt{3}$? Between which two consecutive integer numbers is $\sqrt{2}+\sqrt{3}$?
My thoughts: $\sqrt{2}$ is $\approx1,4$ and $\sqrt{3}$ is $\approx{1,7}$ so their sum must be of the interval $(3;4)$. Any more strict approaches? Thank you in advance!
 [/INST] First we have $\sqrt{2} + \sqrt{3} < \sqrt{4} + \sqrt{4} = 4$. To show that $\sqrt{2} + \sqrt{3} > 3$, we square both sides, and see that it suffices to show that $5 + 2\sqrt{6} > 9$. But this follows from $\sqrt{6} > 2$.
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