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**1. Let m, n, p, q [itex]\in[/itex] Z**

If 0 < m < n and 0< p [itex]\leq[/itex] q, then mp < nq

If 0 < m < n and 0< p [itex]\leq[/itex] q, then mp < nq

**2. Propositions/axioms I can use that relate to inequalities**

**2.4**Let m,n,p [itex]\in[/itex]

**Z**. If m < n and n < p, then m < p

**2.5**For each n [itex]\in[/itex]

**N**there exists an m [itex]\in[/itex]

**N**such that m > n

**2.6**Let m,n [itex]\in[/itex]

**Z**. If m [itex]\leq[/itex] n [itex]\leq[/itex] m then m=n

**2.7 i)**If m < n, then m+p < n+p

**2.7 ii)**If m < n and p < q then m+p < n+q

## The Attempt at a Solution

So far, my first idea is to say if m < n, then sm < sn (s being an arbitrary integer). However, I don't have an axiom or proposition to reference for this step, so I don't even know if I can use it. If I could use it, then i would go on to say that if sm < sn, then if p<q, then pm<qn, but this is the thing they want me to prove! So I can't reference the proposition I am trying to prove, I am just stuck!

The other issue is that I can only use axioms an propositions from earlier in the book, so my options are limited, but the most obviously useful ones seem to be 2.7 (i) and (ii)

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