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For any sequence of decimal digits $x_1x_2 \ldots x_m$ there exists $n \in \mathbb{N} $ such that this seqence occurs (as a substring) in the decimal expansion of the fractional part of $ \sqrt{n} $ . Is it true?

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Moreover, there exists $n\in\mathbb{N}$ such, that $x_1,...,x_m$ are first fractional digits in decimal representation of $\sqrt{n}$. Let $ x=0,x_1x_2...x_m$, so for all such $x$ there exists integer $i$ (big enough): $$(i+x+10^{-(m+1)})^2-(i+x)^2 = 10^{-2(m+1)}+2x10^{-(m+1)}+2i10^{-(m+1)}> 1.$$ So, since $f(y)=(i+x+y)^2$ is continuous function, there exists $y\in[0,10^{-(m+1)}]$ such, that the number $n=(i+x+y)^2$ is integer, and $x_1x_2...x_m$ are first fractional digits of $\sqrt{n}$.

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