Download 1.1 Prove that there is no rational number whose square is 12. Proof

1.1 Prove that there is no rational number whose square is 12.
Rudin’s Ex. 2
Proof Suppose there is a rational number whose square is 12. Hence there are two
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integers m and n, being co-prime, such that m
= 12. This gives m2 = 12n2 ,
n
which implies that m is of multiple of 3. Put m = 3k. Then we have 4n2 = 3k 2 .
This implies that n is also of multiple of 3. Both m and n being of multiple of 3
contradicts to the assumption that they are co-prime.
1.2 Let E be a nonempty subset of an ordered set; suppose α is a lower bound of E and
β is an upper bound of E. prove that α ≤ β.
Rudin’s Ex. 4
Proof Since α a lower bound of E, and E be nonempty, there is x ∈ E such that
α ≤ x. Since β is an upper bound of E, then for this element x, we have x ≤ β. By
the transitivity, we have α ≤ β.
1.3 Let A be a nonempty set of real numbers which is bounded below. Let −A be the
set of all numbers −x, where x ∈ A. Prove that
inf A = − sup(−A).
Proof Since A is nonempty and bounded below, −A is nonempty and bounded above.
By the least-upper-bound-property of R, b = sup(−A) exists. We shall show that
−b = inf(A).
First, for any x ∈ A, since b = sup(−A) is an upper bound of −A, we have −x ≤ b,
or −b ≤ x. This means that −b is a lower bound of A. To show that −b is the
greatest lower bound of A, for any c satisfying −b < c, we only need to show that c
cannot be a lower bound of A. Indeed, Since −c < b, so −c is not an upper bound
of −A. Hence there is x ∈ −A such that −c < x. In other words, we have −x ∈ A
such that −x < c. So c is not a lower bound of A.
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Rudin’s Ex. 5