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**redsoxfan325** If you take the decimal expansion of any irrational number less than one, i.e. $\displaystyle 0.a_1a_2a_3a_4...$ and any finite string of digits $\displaystyle n_0n_1n_2...n_m$ (i.e. a natural number), will you always find that exact string somewhere in the decimal expansion of the irrational number. In other words, does there exist $\displaystyle N$ such that $\displaystyle a_{N+k}=n_k$ for all $\displaystyle k\in[0,m]$?

For instance, say you are given the irrational number $\displaystyle \sqrt{2}-1$ and a string of numbers $\displaystyle 1356237$. Then you have $\displaystyle \sqrt{2}-1=0.4142\textbf{1356237}31...$

Intuitively, it seems like you should be able to, but I have no idea how to justify that claim.

Any ideas?