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

Review Exercises
Discuss the truth of the following statements. Prove the ones that are
true; find a counterexample for each one of the false statements. Exercises
with the symbol (*) require knowledge of calculus or linear algebra.
1. If P(xi,yi) and Q(x2,y2) are two distinct points in the plane, then the
distance between the two of them, defined as:
d(P, Q) =
y/(x2-x,f-(y2-yif
is a positive number.
2. Let a be a real number. Then the opposite of a is unique.
3. If n is any positive integer number, then In n < n.
Prove this statement in all of the following ways:
(a) By induction.
(b) By graphing the two functions f(x) = In x and g{x) = x and
comparing them. Use only x > 1 as the statement is only about
positive integers.
(c) By studying the function h{x) = Inx/x for x > 1. (*)
(Show that the function is bounded by 0 and 1.)
(d) By studying the function ^(x) = Inx — x for x > 1. (*)
(Show that ^(x)<0 for all x > 1.)
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