# Proving that $u_n$ is arithmetic sequence

Let $u_n$ be a sequence defined on natural numbers (the first term is $u_0$) and the terms are natural numbers ($u_n\in \mathbb{N}$ )

We defined the following sequences:

$$\displaystyle \large x_n=u_{u_n}$$ $$\displaystyle \large y_n=u_{u_n}+1$$

If we know that both $y_n$ and $x_n$ are arithmetic sequences ,how we can prove that $u_n$ is also arithmetic sequence

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artofproblemsolving.com/Forum/… – only Aug 28 '12 at 8:17
As written, the fact that $\{y_n\}$ is arithmetic is redundant given that $\{x_n\}$ is. Did you mean: $\displaystyle \large x_n=u_{u_n}$ AND either $\displaystyle \large y_n=x_{u_n}+1$ or $\displaystyle \large y_n=u_{x_n}+1$? – Marconius Jul 9 '15 at 19:51