A pair of mutually "bounding" sequences. Given $({a_n})_{n=1}^{\infty}$, $({b_n})_{n=1}^{\infty}$ convergent sequences
and where
$$\{n\in\mathbb{N}\mid a_n\le b_n\}\quad\text{and}\quad\{n\in\mathbb{N}\mid b_n\le a_n\}$$
are both unbounded, prove that 
$$\lim \limits_{n\to \infty}a_n=\lim \limits_{n\to \infty}b_n$$
I would like to know how I can prove it using simple calculus theorem(I only know the definition of limit, arithmetics of limits and the Squeeze Theorem). 
Thank you very much for your time and help.

If you want to proceed from basics, meaning the $\epsilon$-$N$ definition of limit,  let the limits of our sequences be $a$ and $b$ respectively. 
We show that we cannot have $a<b$, and that we cannot have $a>b$. To show that $a<b$ is not possible, let $\epsilon=(b-a)/3$.  There is an $N$ such that if $n >N$ then $a_n$ is within $\epsilon$ of $a$, and $b_n$ is within $\epsilon$ of $b$. But then we cannot have $a_n \ge b_n$, contradicting the fact that the set of $n$ such that $a_n \ge b_n$ is unbounded.
