This is one of the example problem that has been solved in Velleman's How to prove book:
Find the smallest set of real numbers X such that $5 \in X$ and for all real numbers $x$ and $y$, if $x \in X$ and $x < y$ then $y \in X$.
The solution is $A = \{y \in R \mid 5 \leq y \}$
Although when I think I feel that $\{5\}$ should be the smallest set. Any explanation of how this is solved ?
Let $X$ a set which satisfies this properties and $y>5$. $5\in X$ and $5<y$, ergo $y\in X$. On the other hand we supposed as we used it $5 \in X$. It means $X$ must contain the closed half line $[5,+\infty[$. Our last step is to check that this set satisfies our requirements.