Definition) An event $e$ occurs with high probability if $\Pr(e)>1-\frac{1}{n}$.
Tip) $\big(O(\log{n})\big)! > n $
Let $\sigma = r_{1},r_{2},...,r_{n}$ be a sequence, such that $r_{i}$ $(1\leq i\leq n)$ is random number $\in R$.
What is the length of the longest strictly increasing sub-sequence in $\sigma$ with high probability?
Let $l$ be the length of the longest strictly increasing sub-sequence in $\sigma$. Therefore, there exists at least one sub-sequence $\sigma' = r_{j+1},r_{j+2},...,r_{j+l}$, such that $r_{j+1}<r_{j+2}<...<r_{j+l}$ or $r_{j+1}>r_{j+2}>...>r_{j+l}$. The probability of occurring $\sigma'$ is $\frac{2}{l!}$.
Set $l=O(\log{n})$. Then, $\Pr\{\text{occuring }\sigma'\} = \frac{2}{\big(O(\log{n})\big)!}<\frac{1}{n}$. Therefore, $\sigma'$ does not occur with high probability, so the length of the lonest strictly increasing sub-sequence must be less than $l=O(\log{n})$.
Is my proof correct?