Let $A=(2014,2016]$. Which of the following statements is/are true?
(a) $\sup A=2016$ and $\inf A=2014$
(b) $\sup A=2014$ and $\inf A=2016$
(c) $\sup A=2016$ and $\inf A$ does not exist
(d) $\sup A$ does not exist and $\inf A=2014$
(e) Both $\sup A$ and $\inf A$ do not exist.
My try
Since there is no possible least number inf does not exist and $\sup(A)=2016$.
Is my argument correct?
Best Answer
You are confusing between infimum and minimum. Infimum is the largest lower bound of the set. So the definition doesn't say the infimum must belong to the set itself. If it does belong to the set then it is called minimum. So what is the greatest lower bound of your set? Obviously it is $2014$. Why? First of all it is a lower bound. And it is easy to see that any bigger number than $2014$ is already not a lower bound of $A$.