For reference I am currently taking Introduction to Abstract Math, and have taken Calculus 1, Discrete Math, and Linear Algebra.
Given $(a_n),(b_n),(m_n)$ defined in this post, how would I find the supremum of $(a_n)$ and $(b_n)$, knowing that $\forall n \geq 1, (a_n < \sqrt{2} < b_n = a_n + 2^{-n})$? I am not allowed to use the formal definition of convergence. I have already proved each sequence is monotonic, and that $(a_n)$ is increasing and that $(b_n)$ is decreasing on my own. I am not sure how to go about solving this, given how complicated each of the sequences are. How would I start solving this kind of problem?
HINT: For each $n\ge 1$ you know that $a_n<\sqrt2<a_n+2^{-n}$. Clearly $\sup_na_n\le\sqrt2$, so the real question is whether $\sup_na_n$ could be strictly less than $\sqrt2$. Suppose that $\sup_na_n=s<\sqrt2$. What happens if $n$ is so large that $2^{-n}<\sqrt2-s$?
It’s also clear that $\inf_nb_n\ge\sqrt2$, so for this one you want to ask yourself whether it’s possible for $\inf_nb_n$ to be strictly greater than $\sqrt2$; use the same general idea as I used above.