epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

110 The Nuts and Bolts of Proofs
(b) Is the previous statement true if one considers an infinite sequence of
rational numbers? That is: If ai, a2,,.., a„,... is an infinite sequence
of rational numbers, is the sum:
S = ai + a2 + ... 4- a„ +
a rational number? (*)
00
= X^ ^fc
k=i
39. Let/, g, and h be three functions defined for all real numbers such that
f(x) < h(x) < g{x) for all x. If/and g are decreasing, then h is decreasing.
40. The equation sin x = —x-\- 1/2 has a unique solution for 0 < x < n/l.
41. The product of four consecutive integers increased by 1 is always a
perfect square.
{Hint: Try the square of a trinomial.)
42. Let A and B be two nonempty subsets of the same set U. Then either
BQA' or AnB^&.
43. There exist integer numbers a, b, and c such that be is a multiple of a, but
neither b nor c is a multiple of a.
44. Let n be a natural number larger than 3. Then 2" > n!
45. Let/be a function defined for all real numbers. The function/is even if
and only if its graph is symmetric with respect to the y-Sixis. (A graph is
symmetric with respect to the y-axis if whenever the point (x, y) belongs
to it, the point {—x,y) will belong to it as well.)
46. The systems I . j /• and I . x . /L T r have the
•^ ycx-\-dy=f [(a-c)x-\-(b-d)y = e-~f
same solutions.
{Hint: Prove that {t, s) is a solution of the first system if and only if it is a
solution of the second system.)
47. Let fcbe a natural number. An integer of the form 16/c + 5 is never a
perfect square.
48. Let n be an integer. Then the following four statements are equivalent:
(a) n is odd.
(b) n^ is odd,
(c) (n - if is even.
(d) {n + 1)^ is even.
49. Let/be a positive function defined for all real numbers and never equal
to zero. Then the following statements are equivalent:
(a) /is an increasing function.