6 1. The Real Line
Thus setting 723(e) = max(ni(|a|/2),
ni(a2e/2))
we see from (4) that
I 1 1 |
6
\an a\
for all n 713(e). The result follows.
(vii) The proof of the first sentence in the statement is rather similar to
that of (i). By definition, f holds. Suppose, if possible, that a A, that is,
a A 0. Setting N = n\(a A) we have
&N = (&N a) + a a \ajsj a\ a (a A) = A,
contradicting our hypothesis. The result follows by reductio ad absurdum.
To prove the second sentence in the statement we can either give a similar
argument, or set an —bn, a = —b and A = —B and use the first sentence.
[Your attention is drawn to part (ii) of Exercise 1.8.]
Exercise 1.7. Prove that the first few terms of a sequence do not affect
convergence. Formally, show that if there exists an N such that an = bn for
n N, then an a as n 00 implies bn a as n 00.
Exercise 1.8. In this exercise we work within Q. (The reason for this will
appear in Section 1.5, which deals with the axiom of Archimedes.)
(i) Observe that i / e G Q and e 0; then e = m/N for some strictly pos-
itive integers m and N. Use this fact to show, directly from Definition 1.5,
that (if we work in Q) 1/n * 0 as n 00.
(ii) Show, by means of an example, that, if an a and an b for all
n, it does not follow that a b. (In other words, taking limits may destroy
strict inequality.)
Does it follow that a b? Give reasons.
Exercise 1.9. A more natural way of proving Lemma 1.6 (i) is to split the
argument in two.
(i) Show that if \a b\ e for all e 0, then a = b.
(ii) Show that if an a and an b as n oo; then \a b\ e for all
e 0 .
(Hi) Deduce Lemma 1.6 (i).
(iv) Give a similar 'split proof' for Lemma 1.6 (vii).
Exercise 1.10. Here is another way of proving Lemma 1.6 (v). I do not
claim that it is any simpler, but it introduces a useful idea.
(i) Show from first principles that, if an a, then can ca.
(ii) Show from first principles that, if an a as n 00, then a^
a2.
(Hi) Use the relation xy = ((x +
y)2
(x
y)2)/4
together with (ii), (i)
and Lemma 1.6 (iv) to prove Lemma 1.6 (v).
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