Proof or counterexample : Supremum and infimum If $($An$)_{n \in N}$ are sets such that each $A_n$ has a supremum and $∩_{n \in N}$$A_n$ $\neq$ $\emptyset$ , then $∩_{n \in N}$$A_n$ has a supremum.
How to Prove This.

Let $x \in \cap_{n \in \mathbb N} A_n$. Since $x$ belongs to each $A_n$, $x \le sup\ A_n$ for any $n \in \mathbb N$. Then we see that $\{x : x \in \cap_{n \in \mathbb N} A_n\}$ is bounded above (by $sup\ A_1$ for example) and so by the Completeness Property of $\mathbb R$, $\cap_{n \in \mathbb N} A_n$ has a supremum.
