Use only Peanos axioms Show how to define the product nm of

Use only Peano’s axioms:

Show how to define the product nm of two natural numbers. Hint: use induction on m.

Use the definition of product you gave in the preceding exercise to prove that if n, m N then n nm.

For any natural number m, we define multiplication x as follows:

Define 0 x m = 0

Now, let n be another natural number, and suppose we have inductively defined n x m.

Then, define (n + +) × m := (n × m) + m

Now, we know that for any natural numbers a, b, c

if a<=b

then ac<=bc

Therefore, for a=1, b=m and c=n where m, n are natural numbers such that a<=b i.e. a<=m

Then, ac<=bc i.e. 1n<=mn

Therefore, for any natural numbers n and m, n<=nm