$x_n\leq y_n\ \forall n \in N \implies \inf x_n \leq \inf y_n$
It is obvious from the definition of infimum and supremum, $\sup x_n \leq y_n$ and $\inf x_n \leq x_n \leq y_n$. However I do not know how to use the definition to prove formally that $\sup x_n \leq \inf y_n$ and conclude that
$\inf x_n \leq \sup y_n$.
Best Answer
Note that $$ \inf x_n\leq x_n\le y_n $$ for all $n$. So $\inf x_n$ is a lower bound for $y_n$ whence $$ \inf x_n\leq \inf y_n. $$