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how do I show that every convergent periodic sequence of Real numbers is constant?I have the intuition that is true,but I don't know how to prove.thanks.

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HINT: Let $\langle a_k:k\in\mathbb{N}\rangle$ be your sequence. Suppose that it has period $p$ and that $a_n\ne a_m$. What can you say about the subsequences $\langle a_{n+kp}:k\in\mathbb{N}\rangle$ and $\langle a_{m+kp}:k\in\mathbb{N}\rangle$?

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they are constant, and have different limits, thus the sequence would have 2 accumulation points, a contradiction.Thanks. –  Jr. Jan 13 '12 at 0:47
    
@Jr.: Yep, you’ve got it. –  Brian M. Scott Jan 13 '12 at 0:54
    
Another one: If a monotone sequence has convergent subsequence,prove that the sequence itself is convergent. –  Jr. Jan 13 '12 at 1:01
    
@Jr.: Let $L$ be the limit of the convergent subsequence $\langle a_{n_k}:k\in\mathbb{N}\rangle$. Assume without loss of generality that the sequence is non-decreasing. Let $\epsilon>0$; there is an $m$ such that $a_{n_k}\in(L-\epsilon,L]$ whenever $k\ge m$. What can you say about $a_k$ when $k\ge n_m$? Use monotonicity. –  Brian M. Scott Jan 13 '12 at 1:07

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