Given $\displaystyle\lim_{n\rightarrow \infty} a_{n}=a$ and $\displaystyle \lim_{n\rightarrow\infty} b_{n}=b$, and we have $a < b$, how does one prove that there exist a natural number $K$ such that $a_{n} < b_{n}$ for all $n\geq K$?
I can see that intuitively as the sequence $a_{n}$ approaches its limit, it becomes really close to $a$ and logically that implies that eventually the value of $a_{n}$ will be smaller than $b_{n}$ when $n$ is sufficiently large, however I can't seem to find a mathematically sound argument to prove my point. I'm thinking if it's possible to solve this question using the epsilon-K thingy, but I kinda have no idea how to start on my proof. Could someone point me in the right direction?