If $\{\max(a_n,b_n)\}_{n\in \mathbb{N}}$ and $\{\min(a_n,b_n)\}_{n\in \mathbb{N}}$ converge, do $\{a_n\}_{n\in \mathbb{N}}$ and $\{b_n\}_{n\in \mathbb{N}}$ converge?
An observation: the converse is true. My attempt to solve question: It is obvious that $\min(a_n,b_n) \leq a_n \leq \max(a_n,b_n)$ and $\min(a_n,b_n) \leq b_n \leq \max(a_n,b_n)$ for all $n\in \mathbb{N}$, so if $\{\max(a_n,b_n)\}_{n\in \mathbb{N}}$ and $\{\min(a_n,b_n)\}_{n\in \mathbb{N}}$ converge to the same limit, so do $\{a_n\}_{n\in \mathbb{N}}$ and $\{b_n\}_{n\in \mathbb{N}}$. I suspect this doesn't hold when $\{\max(a_n,b_n)\}_{n\in \mathbb{N}}$ and $\{\min(a_n,b_n)\}_{n\in \mathbb{N}}$ converge to different limits, but I cannot provide a counterexample.
Take two numbers $\alpha$ and $\beta$ and define$$a_n=\begin{cases}\alpha&\text{ if $n$ is odd}\\\beta&\text{ if $n$ is even}\end{cases}\quad\text{and}\quad b_n=\begin{cases}\beta&\text{ if $n$ is odd}\\\alpha&\text{ if $n$ is even.}\end{cases}$$Then$$\lim_{n\to\infty}\max\{a_n,b_n\}=\max\{\alpha,\beta\}\quad\text{and}\quad\lim_{n\to\infty}\min\{a_n,b_n\}=\min\{\alpha,\beta\}.$$But, unless $\alpha=\beta$, the limits $\lim_{n\to\infty}a_n$ and $\lim_{n\to\infty}b_n$ do not exist.