220 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6
(c) The additive inverse axiom for R states that, corresponding to every x E R, there
is a real number y such that x + y = y + x = 0. Prove that this y is uniquely
determined by x. (Use the fact that, for any real numbers a, b, and c, if a = b, then
c + a = c + b.) We denote this unique value of y by -x.
(d) The multiplicative inverse axiom for R states that, corresponding to any non-
zero x E R, there is a real number y such that xy = yx = 1. Prove that this y is
uniquely determined by the given nonzero x. (Use the fact that, for any real
numbers a, b, and c, if a = b, then ca = cb.) We denote this unique value of y
by x-'.
- (a) Prove that 0 is the only real number x satisfying the statement ax = x for all
a E R. [Assume the theorem a - 0 = 0 for all a E R. (Hint: Use specialization.)]
*(b) Prove that 0 is the only subset X of a universal set U satisfying the state-
ment A n X = X for all sets A s U. (Assume that A n (25 = 0 for all sets
A c U.)
(c) Prove that^0 is the only subset X of a universal set U satisfying the state-
ment A u X = A for all sets A = U. (Assume that A u 0 = A for all sets
A E U.)
- (a) In Example 3, Article 6.1, we defined invertibility for an n x n matrix A.
Show that if an n x n matrix A is invertible, then an associated matrix B such
that AB = BA = I, is unique. We denote this uniquely determined matrix by
A- I.
(b) Prove that if A = (aij), x2 has a,,a,, - a,,a,, # 0, then A is invertible.
(c) Prove that the 2 x 2 matrix (i is not invertible.
- (a) Give an indirect proof of the uniqueness of lim,,, f(x), where f is defined
in an open interval containing a. That is, prove that if L, satisfies the epsilon-
delta definition of L = lim,,, f (x), and L, # L,, then L, cannot satisfy this
definition.
(b) Recall from Example 5, Article 6.1 the definition of x = limn,, x,. Mimic the
proof given in Example 5 to show that a limit of a convergent sequence is unique.
*(c) Recall from Exercise 21, Article 6.1, the definition of cluster point of a sequence.
Prove that if the sequence {x,) converges to the real number x, then this x is the
unique cluster point of the sequence.
- (a) Prove that the set {llnl n = 1,2,3,.. .) is bounded above in R. (cf., Ex-
.. -- - ample 7).
(b) A subset S of R is said to be bounded below in R if and only if there exists a real
number L such that L I x for all x E S. Prove that if S is bounded above in R,
then the set -S = {x E RI -x E S) is bounded below in R.
*(c) Prove that if S, and S, are both bounded above in R, then S, u S, is bounded
above in R.
(d) A subset S of R is said to be bounded in R if and only if there exists M > 0
such that 1x1 5 M for all x E S. Prove that S is bounded in R if and only if S is
bounded above and bounded below in R.
(e) Prove that if (x,) is a convergent sequence of real numbers, then the set
{x,, x,,.. .) is a bounded set.
- (a) Let f (x) = sin x and A = [- n/4,7~/4]. Prove that a E f (A).
