By Alain Guichardet
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Extra resources for Symmetric Hilbert spaces and related topics; infinitely divisible positive definite functions, continuous products and tensor products, Gaussian and Poissonian stochastic processes
4 are still satisﬁed for this set. 5. But it is diﬃcult to quantify what we mean by “can be obtained from” without already using the natural numbers, which we are trying to deﬁne. 5 (Principle of mathematical induction). Let P (n) be any property pertaining to a natural number n. Suppose that P (0) is true, and suppose that whenever P (n) is true, P (n++) is also true. Then P (n) is true for every natural number n. 10. We are a little vague on what “property” means at this point, but some possible examples of P (n) might be “n is even”; “n is equal to 3”; “n solves the equation (n + 1)2 = n2 + 2n + 1”; and so forth.
Proof. We shall just prove the ﬁrst claim. Suppose that A ⊆ B and B ⊆ C. To prove that A ⊆ C, we have to prove that every element of A is an element of C. So, let us pick an arbitrary element x of A. Then, since A ⊆ B, x must then be an element of B. But then since B ⊆ C, x is an element of C. Thus every element of A is indeed an element of C, as claimed. 19. 7. 40 3. 20. There is one important diﬀerence between the subset relation and the less than relation <. 13); however, given two distinct sets, it is not in general true that one of them is a subset of the other.
Thus the base case is done. Now suppose inductively that n + m = m + n, now we have to prove that (n++) + m = m + (n++) to close the induction. By the deﬁnition of addition, (n++) + m = (n + m)++. 3, m + (n++) = (m + n)++, but this is equal to (n + m)++ by the inductive hypothesis n + m = m + n. Thus (n++) + m = m + (n++) and we have closed the induction. 5 (Addition is associative). For any natural numbers a, b, c, we have (a + b) + c = a + (b + c). Proof. 1. Because of this associativity we can write sums such as a + b + c without having to worry about which order the numbers are being added together.
Symmetric Hilbert spaces and related topics; infinitely divisible positive definite functions, continuous products and tensor products, Gaussian and Poissonian stochastic processes by Alain Guichardet