Prove that $\varlimsup_{n\to\infty}(\max\{x_n,y_n\})=\max\{\varlimsup_{n\to\infty}x_n,\,\varlimsup_{n\to\infty}y_n\}$

Question

Prove, that $\varlimsup_{n \to \infty}(\max\{x_n, y_n\}) = \max\{\varlimsup_{n \to \infty} x_n,\, \varlimsup_{n \to \infty} y_n\}$. Let $(x_n)$ and $(y_n)$ be bounded number sequences. Does also the following equality always stand $\varlimsup_{n \to \infty} (\min\{x_n, y_n\}) = \min\{\varlimsup_{n \to \infty} x_n, \,\varlimsup_{n \to \infty}y_n\}$? In the proof, should I use upper limit monotonicity and/or upper limit description with partial sequences? Any tips or advice? I'm a little stuck and don't know where to begin.

Answer 1

You can write the max-function as $\frac{x_{n}+y_{n}+|x_{n}-y_{n}|}{2}$ . Show continuity and then u can pull limes into the max-function.