1549901369-Elements_of_Real_Analysis__Denlinger_

2.6 Subsequences and Cluster Points 109

Proof. (a) This is merely a restatement of the definition of Xn---+ L.
(b) Part 1 (=?):Suppose {xn} h as a subsequence {xnk} converging to L.
Let c: > 0. Since Xnk ---+ L , 3 ko E N 3 k ;::: ko =? lxnk - LI < c:. Then infinitely
many terms of { Xnk} are in ( L - c:, L + c:); hence, infinitely many terms of { Xn}
are in ( L - c:, L + c:).

Part 2 (~):Suppose {xn} is a sequence 3 Ve:> 0 , {xn} is frequently in
( L - c:, L + c:). We define a subsequence { Xnk} by the alternate principle of
mathematical induction on k (Theorem 1.3.9):

(1) Since { Xn} is frequently in (L - 1, L + 1), 3 n 1 E N 3 Xn, E

(L - 1, L + 1). This defines Xn,.

(2) Suppose we have found Xn,, Xn 2 , • • · , Xnk 3 Vi = 1, 2, · · · , k,

(

1 1).

Xn, E L - --:, L +--: and ni < n2 < · · · < nk· We shall find Xnk+i ·

i i

Since {xn} is frequently in (L - -k

1

, L + -k

1

) , 3 infinitely

" + 1 + 1

many natural numbers n such that Xn E ( L - k!

1

, L + k!

1

) ·

Hence, 3 nk+l > nk 3 Xnk+i E (L --k

1

, L + -k

1

).

·+1 +1

Hence, by the alternate principle of mathematical induction, Vk E N,

3 xnk E (L-~,L+ ~),and ni < n2 < · · · < nk < nk+l <···.Hen ce,

{ Xnk} is a subsequence of { Xn} and by the squeeze principle,

lim Xnk = L.

n->oo

Therefore, { Xn} has a subsequence { Xnk} converging to L. •

The following theorem is frequently useful.

Theorem 2.6.8 A sequence {xn} converges to a real number L ¢:? every sub-
sequence of { Xn} converges to L.

Proof. Part 1 ( =?): Suppose Xn ---+ L. Let { Xnk} be any subsequence of
{xn}· We shall prove that Xnk ---+ L.
Let c: > 0. Since Xn---+ L , 3n 0 EN 3 n;::: no=? lxn - LI< c:. Then,

k;::: n 0 =? nk ;::: k;::: n 0 (see Lemma 2.6.3);

=? lxnk - LI < c:.