In $101$ random integer numbers $a[i],i=0, \cdots,100$, prove that we can always find $10$ non-increasing or non-decreasing sequence.
A sequence is a sequence of numbers is an array of numbers picking up from the original array with increasing index.
We say a sequence starts on the left and ends on the right.
This is just the Erdős-Szekeres theorem. Assign to each number $k$ in the sequence a sticker $k_i,k_d$. Where $k_i$ is the length of the longest non-increasing sequence ending at $k$, and $k_d$ is the length of the longest non-decreasing sequence sequence ending at $k$. See that if there are no sequences of length 11 or higher then there are $100$ possible stickers . Now prove that no two elements can have the same sticker and you are done.(by contradiction, because you can elongate the longest sequence of one of the numbers).
Once you have shown no two numbers in the sequence can have the same sticker, since you have 101 numbers then there must be more than 100 stickers, and then you must have a monotonic sequence of length 11 or more.
From that one you can extract a monotonic sequence of size 10.