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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?