2 Hints

As per last week, here’s the hints section of this document.

• [H1.] Recall that a number \(N\) is a multiple of 3 if there exists \(j \in \mathbb{Z}\) such that \(N = 3j\). The proof is then similar to tutorial question 1.
• [H2.] Try and get the problems into the form of one of the order axioms. Make sure to state each axiom you use, when you use it!
• [H3.] The induction should be straightforward. To find the formula you need to prove, have you seen a way of rewriting \(\begin{pmatrix} n\\ 10 \end{pmatrix}\) recently? (Have a look at the proof of the binomial theorem).
• [H4.] In both cases, your contradiction will come from the fact that one number is found to be strictly less than itself.
• [H5.] Think back to the definitions, and use them to construct your proof of this result.
• [H6.] The method here is very similar to that used in proving the infimum result from Tutorial Question 5.