A question about Infimum.

Question

Let $A$ be an infinite set that includes Real numbers and its not 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.

• prove or disprove the existence of $\inf B$ if $A$ is not bounded.

Actually this is the third part of the question that I didn't solve. I think there is more than one situation in this case. can someone give me a hint ?

Answer 1

Consider $A=\{-n:n\in \mathbb{N}\}$ and $B=A.$

Answer 2

Hint: Notice that $A$ being bounded means that there exists $M \in \mathbb{R}$ a positive number s.t. $A \subset (-M,M)$, which implies that $A \cap [M,+\infty)= \emptyset$. Try to solve it from there.