Be Reasonable
Age 16 to 18
Challenge Level 
Try a proof by contradiction. Suppose that the three irrational numbers do occur in some arithmetic series. Can you then go on to reach a contradiction?
It helps to have seen a proof that $\sqrt 2$ is irrational and
to appreciate how the logic of arguments by contradiction
work.
See
Proof Sorter and, for some further reading on proofs by
contradiction, see
this article written by two undergraduates.
Then you only need to know the definition of an arithmetic
series to do this problem. If the difference between $\sqrt 2$ and
$\sqrt 3$ is an integer multiple of the common difference in an
arithmetic series, and the difference between $\sqrt 3$ and $\sqrt
5$ is also an integer multiple of that common difference, can you
use these two facts to write down two expressions, eliminate the
unknown common difference and then find an impossible relationship?
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