Solutions for exercises in chapter 1 E1.1 Prove by induction that for ...

For any positive integer m let g ′ m be the function whose domain is the set of all m-tuples of
real numbers such that g ′ m (〈a0, . . ., am−1〉) = g(〈a0, . . ., am−1〉). We claim that g ′ satisfies
the conditions of the exercise.
for m ≥ 2 and a = 〈a0, . . ., am〉,
g ′ 1(〈a0〉) = g(〈a0〉)
= f(〈a0〉, g〈pred(A, 〈a0〉, R))
= a0;
g ′ m+1 (a) = g(a)
= f(a, g ↾ prec(A, a, R))
= (g ↾ prec(A, a, R))(〈a0, . . ., am−1〉) + am
= g((〈a0, . . ., am−1〉) + am
= g ′ m(〈a0, . . ., am−1〉) + am.
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