6.1 CONCLUSIONS INVOLVING V, FOLLOWED BY 3 203- Prove that lirn,,, f(x) exists if and only if both lim,,,+ f(x) and lim,,,- f(x)
exist and are equal. - (Some prior familiarity with infinite sequences beneficial in Exercises 20,21, and
- A real number x is said to be a limit of a sequence {x,), denoted x = lirn,,, x,
or simply x, -r x (we also say that the sequence x, converges to x) if and only if
VE > 0, 3 a positive integer N such that lx, - XI < E whenever n 2 N. This means
that any E neighborhood of x, no matter how narrow, must contain all but a finite
number (namely, some or all of the first N, where N depends on E) of the terms in
the sequence. The Archimedean property of the real numbers, which we assume,
says in essence that the sequence l/n converges to 0; that is, limn,, l/n = 0 (recall
Example 2 ff., Article 4.2).
(a) Write the logical negation of the definition of sequential convergence; that
is, What is true if x # lirn,,, x,?
n even
(b) Prove that 0 # limn,, x,, where xn =n > 10,000
(c) Prove that^0 = lirn,,, x,, where x, =(d) Prove that, if x, -r x and k is a real number, then kx, + kx.
n even
(e) Prove that^0 = lirn,,, x,, where x, = - 1 n odd '(f) Prove that if x, + x and y, + y, then x, + yn -+ x + y [recall Exercise 15(b)].
- (Continuation of 20) A real number x is said to be a cluster point of a sequence
{x,), if and only if, for all E > 0 and for all positive integers N, there exists n 2 N
such that Ix, - xl < E. This means that any E neighborhood of x, no matter how
narrow, must contain infinitely many terms of the sequence.
(a) Write the logical negation of the definition of cluster point.
(b) Show that 1 is not a cluster point of the sequence {x,) defined by x, = lln.
(c) Show that + 1 and - 1 are both cluster points of the sequence (x,} defined by
x, = (- 1)". Thus cluster points of a sequence are not necessarily unique. (We
will see in Article 6.3 that a limit of a sequence, if it exists, & unique.)
(d) Prove that if lim,, , x, = x, then x is a cluster point of {x,). What does this
say about the relative strength of the properties "x is a cluster point of {x,)"
and "x is a limit of {x,}"?
(e) Prove that 2 is a cluster point of the sequence.{x,) defined by
n, n # 5k for any positive integer k
2, n = 5k for some positive integer k
Does this sequence have any other cluster point(s)? Does it have a limit? - A sequence (x,} is said to be a Cauchy, or fundamental, sequence if and only
if, for all E > 0, there exists N E N such that m 2 N and n 2 N imply Ix, - x,l < E.
(a) Prove that if {x,) converges, then {x,} is Cauchy.
(6) Prove that if {x,) is Cauchy, and if x is a cluster point of {x,), then (x,}
converges to x.