I know $\sqrt2$ and $\sqrt3$ are irrational numbers. However, how can I prove that $\sqrt2$ + $\sqrt3$ is an irrational number? Thank you.
I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. My first question is, is this reasoning correct?  Secondly, the book wants me to use the fact that if $n$ is an integer that is not a perfect square, then $\sqrt n$ is irrational. This means that $\sqrt 6$ is irrational. How are we to use this fact? Can we reason as follows:  $\sqrt 6$ is irrational  $\Rightarrow \sqrt{2 \cdot 3}$ is irrational.  $\Rightarrow \sqrt 2 \cdot \sqrt 3$ is irrational  $\Rightarrow \sqrt 2$ or $\sqrt 3$ or both are irrational.  $\Rightarrow \sqrt 2 + \sqrt 3$ is irrational.  Is this way of reasoning correct?
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