Is there a general way to show that for some sets $A,B$, $\sup(A) = \inf(B)$? I have two sets that satisfy have this property (both sets are in $\mathbb{R}^1$):

$\sup(A) = \inf(B)$

I have troubles to prove it, so I wonder is there a general way to go about  this? I don't want to specify the sets as I want to prove it by myself.
 A: I would say the easiest line of a approach for a problem like this is to look at the definitions of $\sup$ and $\inf$. That is, let $s \ge a$ for all $a \in A$ and if $b$ is any upper bound of $A$, then $s \le b$ implies $s = \sup A.$ And the definition for $\inf B$ is similar. 
A general approach would be to show that $s \in \mathbb R$ that satisfies the two conditions of $\sup A$ also satisfy the two conditions of the definition of $\inf B$.
Without any other details on the problem, this is about all I can suggest.
A: One way to do it :


*

*First prove that $$\forall x \in A, y \in B, x \leq y$$
this will raise $sup(A) \leq inf(B)$

*Then find $(a_n) \in A, (b_n)\in B$ so that $\lim a_n = \lim b_n = l$
You will then have $$l \leq sup(A) \leq inf(B) \leq l$$
and finally $$sup(A) = inf(B)$$
A: You could prove that
$$(\forall (a,b)\in A\times B) \;\;a\le b $$
and
$$(\forall \epsilon>0) \;\;(\exists (a,b)\in A\times B)  :  b-a <\epsilon $$
this is similar to Cauchy criteria for Riemann integrability.
