Question

1) Prove that if the sets S-T and T-S are equivalent, then S and T are equivalent.
2) Prove that
\[ sup\{r \in Q : r< \sqrt{5} \}=\sqrt{5} \]

Answer 1

i am not sure what the first one means, but the idea for the second one is this:
suppose by way of contradiction that the supremum is less than \(\sqrt{5}\) say it is \(\sqrt{5}-\epsilon\)
then since we know that between any two reals there is a rational, there exist some rational \(r\) with between \(\sqrt{5}-\epsilon<r<\sqrt{5}\) contradicting the assumption that \(\sqrt{5}-\epsilon\) is the supremum)

Answer 2

thank you satellite, this helped me a lot..