Prove / disprove:
If $A,B \subset (0,1)$ not empty, and $\inf B = \sup A =: x$ then $A \cap B = \{x\}$
I first tried to disprove by giving a counter-example such as
$$ A = \{ \frac1n | n \in \mathbb{N} \setminus \{1\}\} \\ B = ? \\ \sup A = \frac12$$
But then I wanted to say $B = \{\frac12\}$ this actually satisfies the sentence.
So then I moved to prove this statement, but to no avail, I would appreciate if you could help me!
Assuming $\sup A = x$ then $\forall_{n \in A} ( n < x)$ and $\inf B = x$ then $\forall_{m \in B} ( m > x)$ if $A \cap B = \emptyset $ it is not possible (why?) I really have no clue and I am just talking gibberish..