Question about Infimum. (conclusion about intersection)

Let $A$ be an infinite set that includes Real numbers and is bounded. Let $B$ be a set of Real numbers $x$ s.t. the intersection $A\cap[x,\infty)$ is empty or includes $finite$ number of elements.

  1. prove that $\inf B$ exists.

  2. prove or disprove $\inf B=\min B$

  3. prove or disprove that $\inf B$ exists if we don’t ask for $A$ to be bounded.

I started searching for examples to see the whole picture, like sequences of the form “$\frac{1}{n}”$ and others, but I didn’t know to to do the intersection and why I must be sure that B has an infimum.

its easier if someone can give an example of two sets $A$ and $B$, and make it clear why the intersection must be a finite set. (after choosing “x” from set B)

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