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I am trying to prepare for a quiz and am really stuck on this question. Any help with this would be greatly appreciated!

Let $\{x_n\}$ be a sequence of real numbers and let $y_n = \max \{x_1, x_2, \ldots , x_n\}$ for each positive integer $n$.

Suppose that $\{x_n\}$ is not bounded above. Prove that $\{y_n\}$ converges to infnity.

I am really at a loss on how to prove this.

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  • $\begingroup$ yes, you're correct. I'll change that. $\endgroup$ – JxxYsde3 Feb 26 '17 at 20:33
  • $\begingroup$ $\lim y_n=\limsup x_n$. $\endgroup$ – Nosrati Feb 26 '17 at 20:35
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    $\begingroup$ @MyGlasses: that's not true in general: suppose that $x_n=\frac{1}{n}$. $\endgroup$ – carmichael561 Feb 26 '17 at 20:37
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Hint

You should note (and prove ...) that the sequence $(y_n)$ is increasing.

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If $\{x_n\}$ is not bounded above, then for every real number $M>0$ there exists a natural number $N$ such that $x_N>M$. But then this implies that if $n\geq N$ then $$ y_n=\max\{x_1,\dots,x_n\}\geq x_N>M$$ and since $M$ was any positive real number this shows that $y_n\to\infty$.

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