If $\lim_{n \to \infty} a_{n} = a$ and $\lim_{n \to \infty} b_{n} = b$, how can we show that $\lim_{n \to \infty} \min\{a_{n},b_{n}\} = \min\{a,b\}$?
I say $\min\{a_{n},b_{n}\} $ has two cases: $a_{n}$ and $b_{n}$. So (1) $\lim_{n \to \infty} a_{n} = a$ and (2) $\lim_{n \to \infty} b_{n} = b$. Now I don't know how to imply the $\min\{a,b\}$.