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How can I prove that the set of numbers with two different decimal expansions is countable.

(for example 0.5 and 0.49999...)

This is really hard for me, I tried to prove that this expansions can be as a rational and then since the rationals are countable, I prove this but someone can help please.

Thanks for your time and help.

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    Any such number is equal to $\pm (A+B\cdot 10^{-n})$ for some non-negative integers $ A,B$ so it is rational. – 2017-02-26

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The only time a number has two such representations is when one representation ends in 9999..., preceded by a finite number of decimals, meaning that the other representation has to have a finite number of decimals, and is therefore rational. So yes, you are right, you are dealing with rational numbers here!