If a sequence ($a_n$) is monotonically increasing, and ($b_n$) is a decreasing sequence, with $\lim_{n\to\infty}\,(b_n-a_n)=0$, show that $\lim a_n$ and $\lim b_n$ both exist, and that $\lim a_n=\lim b_n$.
My attempt:
To show that the limits of both sequences exist, I think I should be using the Monotone Convergence Theorem (MCT). For that I would need to show that the sequences are bounded.
($a_n$) is increasing, and so it should be bounded below. ($b_n$) is decreasing, so it should be bounded above. The challenge here is to show that ($a_n$) can be bounded above and ($b_n$) can be bounded below. This should utilise the third condition, from which I get:
$$\begin{align*} & \lim_{n\to\infty}\,(b_n-a_n)=0 \\ \iff & \forall\epsilon>0, \exists N\in \mathbb{N} \text{ s.t. } \forall n\geq N, |{b_n-a_n}|<\epsilon \end{align*}$$
I then tried using the triangle inequality: $$ |b_n|-|a_n|\leq|b_n-a_n|<\epsilon$$
but I'm not sure where to go from here.
