2.6 Subsequences and Cluster Points 107( c) If { Xn} and {yn} are sequences of nonnegative real numbers that are
monotone increasing, then so is their product {XnYn}·
(d) If {xn} and {yn} are arbitrary sequences of real numbers that are
monotone increasing, then so is their product {XnYn}·
(e) If {xn} and {Yn} are sequences of positive real numbers that aremonotone increasing, then so is their quotient { ~= }.
24. Prove that if { Xn} is monotone, then { Xn} converges iff { x~} converges.
What if {xn} is not monotone?- Prove that the sequence J2, J2 + J2, V2 + J2 + J2,
J2 + V2 + J2 + J2, · · · converges and find its limit. What happens
when all the 2's are replaced by 3's?- Let a be a fixed positive real number. Prove that the sequence Jo,, J a + Ja,
Va + J a + Ja, J a + Va + J a + Jo,, · · · converges, and find its limit.
[Suggestion: When proving the sequence is bounded, consider two sepa-
rate cases: a 2 2 and 0 < a < 2.J Find the first five integer values of a for
which the limit L is an integer. What do you notice about these limits?
Conjecture the next integer value of a for which L is an integer. Write L
as a function of a. How does this help you generate more integer values
of a and L that "work?"2.6 Subsequences and Cluster Points
Intuitively speaking, when some of the terms of a sequence are deleted from it,
a new sequence results, which we call a subsequence of the original sequence.
We make this idea rigorous in the following definition.
Definit ion 2.6.1 Suppose {xn} is a sequence. If {nk} is a strictly increasing
sequence of natural numbers; (i. e ., n 1 < n2 < · · · < nk < · · · ) then the sequence
{ Xnk} is said to be a subsequence of { Xn}. Thus, { Xnk} is the sequence
Examples 2.6.2 The following are examples of subsequences of {xn}:
X1, X3, X5, X7, · · · X2n-l1 · · ·
X2, X4, X5, Xs, ... X2n, ...
X5, X10, X15, X20, ... Xsn,...
Xs, Xg, X10, X11, ... Xn+71 ....