Answer to Question #146710 in Real Analysis for Dolly

Question

If A={x∈Q,x<3} then find Sup(A)

Answer 1

Claim: $\sup(A)=3$

Proof:

Clearly, by definition for all $x\in A$ , $x<\sup(A)=3$

Now, for every $\epsilon:=3-\frac{x}{2}>0$ , $3-\epsilon=\frac{x}{2}<x$ $<3$, $3-\epsilon$ is not an upper bound of $A$. Hence done.