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

160 The Nuts and Bolts of Proof, Third Edition
Thus
lim \f{x) -m] = 0
and
lim/(x) =/(a).
x-^a
18. (This statement can be rewritten as: Let p be a number larger than 2.
If p is prime, then p is odd. Because there are infinitely many prime
numbers larger than 2, we cannot check directly that they are indeed
all odd numbers.) Let p be a number larger than 2. We will assume
that p is not odd. Then p must be even. Thus, p = 2n where n is some
natural number. Therefore, 2 is a divisor of p, and 2/p. This
contradicts the fact that p is a prime number. (Be careful. Not all odd
numbers larger than 2 are prime.)
19. We can show that statement 1 is equivalent to statement 2, and that
statement 2 is equivalent to statement 3. Thus, the proof will have
four parts:
Part 1: Statement 1 implies statement 2. Let A~^ be the inverse
of the matrix A. Let /2x2 be the 2x2 identity matrix
(i-e., hxi = I A 1 I )• Then Ayi A = 12x2- By the properties of the
determinant det(^ x yl) = det^ x det^~^ = det/2x2 = L Therefore,
det^T^O.
Part 2: Statement 2 impHes statement 1. We will explicitly find the
matrix A~^ using the coefficients of the matrix A. Let
^=(: 2) "^'-'-{: i>
We want to construct A~^ such thatv4 x A = hxi- From this
we obtain a system with four equations in the four unknowns x, y,
z, and t.
ax-\-bz = I
CX + dz = 0
ay-^bt = 0
cy-\-dt= 1.