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To prove the if part of the statement, let $x_n$ be a constant sequence. That is, $x_n = L$ for some $L\in\mathbb{R}$ for all $n\in\mathbb{N}$. This implies that $x_n \leq x_{n+1}$ for any $n$ as well as $x_n \geq x_{n+1}$. Hence, $x_n$ must be both monotone decreasing and monotone increasing.

To prove the only if part, let $x_n$ be both monotone increasing and monotone decreasing. That is, $x_n \leq x_{n+1}$ and $x_{n} \geq x_{n+1}$ for any $n\in\mathbb{N}$ which in turn implies $x_n = x_{n+1}$. Hence, $x_n$ must be a constant sequence. $\blacksquare$

Although I am convinced my proof is correct most of the times, I am never $100$% sure that my proof is correct. What would be the good practice to verify my proof is correct besides asking my professor or on math.stackexchange?

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    $\begingroup$ Your proof is correct. But at the end, you are asking a tricky question :) $\endgroup$ – user 1987 May 25 '17 at 1:36
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    $\begingroup$ Usually, if you are not sure there are some parts on your proof which you are doubting, so one thing to do is include more details for your arguments until your doubt fades away $\endgroup$ – clark May 25 '17 at 1:49
  • $\begingroup$ @clark can you explain more, I didn't get it? $\endgroup$ – Red shoes May 25 '17 at 7:17

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