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

Exercises Without Solutions 109
22. Prove that if ^= (J | ), then yl"= K J j for all n>2. (*)
23. Let a and b be two integers such that 6 is a multiple of a. Then b^ is a
multiple of a^ for every natural number k.
24. The squares of two consecutive integers are not consecutive integers.
25. Let X and y be two negative numbers. Show that the following four
statements are equivalent:
(a) \x\<\y\
(b)0<^<l
(c) 1<^
(d) x^ < y^
26. An integer is a multiple of 5 if and only if its unit digit is either a 5 or a 0.
27. The squares of two consecutive positive integers are not consecutive
positive integers.
28. Let / be a function defined for all real numbers. The function / is
ffc) -fix)
decreasing if and only if < 0 whenever x / c.
c — X
29. The difference of two irrational numbers is always an irrational number.
30. Let/(x) = 4 ^ - Then lim/(x) = 1/6.
31. If is an integer with n > 1, then (n + 1)^ > Irp-.
32. Let fl be a positive number. Then a < 1 if and only if a^ < a.
33. Let n be a nonnegative integer. The number 10^""^^ + 1 is divisible by 11.
34. The set A of all odd integer multiples of 3 coincides with the set
B = {n = 6k -\- 3 where k is an integer}.
35. Let/ and g be two functions defined for all real numbers and such that
the function f o g is well-defined for all real numbers. If / is a
differentiable function and ^ is a nondifferentiable function, then fog
is nondifferentiable. (*)
36. Let / and g be two increasing functions defined for all real numbers.
Then the function h(x) =f(x)g(x) is an increasing function.
^7 r^ A .u \ 2n+ll^ _ ,. 2
. Then lim «„ = -.
37. Consider the sequence i a„ = -3n + 5
38. (a) If fli, ^2, ... , Un is a finite col lection of rational numbers, then
the sum:
5 = ai + ^2 + ... + fln is a rational number.