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Given the sequence of real numbers $\{x_n\}_{n \in \mathbb N}$, we define $\{y_n\}_{n \in \mathbb N}$ where $y_n=\max\{|x_1|,...,|x_n|\}$ for each $n \in \mathbb N$. Prove that if $\{x_n\}_{n \in \mathbb N}$ is bounded, then $\{y_n\}_{n \in \mathbb N}$ is a convergent sequence.

My attempt at a solution.

If $\{x_n\}_{n \in \mathbb N}$ is bounded, then, it has a convergent subsequence. Call that sequence $\{x_{n_k}\}_{k \in \mathbb N}$. Note that if $x=\lim_{k \to \infty} x_{n_k}$, then $|x|=\lim_{k \to \infty}|x_{n_k}|$. This can be proved by the fact that $0\leq ||x_{n_k}|-|x||\leq |x_{n_k}-x| \to 0$ when $k \to \infty$.

I was going to try to prove that $|x|=lim_{n \to \infty} y_n$ but immediately realize that this doesn't need to be true. For example: $\{x_n\}_{n \in \mathbb N}$: $x_n=0$ if $n$ is odd and $x_n=1$ if $n$ is even has two convergent subsequences.

My problem is I don't know what else to do, I would appreciate some guidance.

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To show that $y_n$ is convergent, focus on the properties of $y_n$. There are several facts about $y_n$ that you should be able to deduce directly from your above statements. Do those facts help you to prove convergence? –  John Dec 20 '13 at 19:40
    
@John right, $\{y_n\}_{n \in \mathbb N}$ is bounded and it is monotone increasing, I can't believe I didn't realize it before –  user100106 Dec 20 '13 at 19:43
    
@user100106 : the MSE system dislikes unanswered questions. You can answer your own question, or John can answer it. –  Stefan Smith Dec 20 '13 at 20:04
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2 Answers 2

up vote 1 down vote accepted

To show that $y_n$ is convergent, focus on the properties of $y_n$. There are several facts about $y_n$ that you should be able to deduce directly from your above statements. Do those facts help you to prove convergence?

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It is consequence of the following:

If a sequence is monotonic and bounded, then it converges.

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As realized by the OP 50 minutes before this was posted. –  Did Dec 20 '13 at 21:42
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