8/16/2019 Real Analysis proof.pdf
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Theorem. (Bolzano-Weierstrass)
Every bounded sequence has a convergent subsequence.
proof:
Let be a bounded sequence. Then, there exists an interval
suchÖA × Ò + ß , Ó8 " "that for all+ Ÿ A Ÿ , 8Þ" 8 8
Either or contains infinitely many of . ThatÒ + ß Ó Ò ß ,
Ó ÖA ×" " 8+ , + ,
# #" " " " terms
is, there exists infinitely many in such that is in or there
exists8 N + Ò + ß Ó8 "+ ,
#" "
infinitely many in such that is in . If contains8 N + Ò ß , Ó Ò
+ ß Ó8 " "+ , + ,
# #" " " "
infinitely many terms of , let . Otherwise,ÖA × Ò + ß , Ó œ Ò +
ß Ó8 "+ ,
#2 2 " "
let Ò + ß , Ó œ Ò ß , ÓÞ2 2+ ,
# "" "
Either or contains infinitely many terms of . If Ò +
ß Ó Ò ß , Ó ÖA ×# # 8+ , + ,
# ## # # #
Ò + ß Ó ÖA × Ò + ß , Ó œ Ò + ß Ó# 8 $ $ #+ , + ,
# ## # # # contains infinitely many terms of , let .
Otherwise, let By mathematical induction, we can continue thisÒ
+ ß , Ó œ Ò ß , ÓÞ$ $ #+ ,
## #
construction and obtain a sequence of intervals a such thatÖÒ ß
, Ó×8 8
) for each a contains infinitely many terms of3 8ß Ò ß ,
Ó ÖA ×ß8 8 8 ) for each a a and 33 8ß Ò ß , Ó © Ò ß , Ó
ß8" 8" 8 8 ) for each b333 8ß + œ † Ð, + ÑÞ8" 8" 8 8
"#
The nested intervals theorem implies that the
intersection of all of the intervals
Ò ß , Ó A ÖA ×a is a single point . We will now construct a
subsequence of which8 8 8will converge to A Þ
Since a contains infinitely many terms of , there exists
inÒ ß , Ó ÖA × 5 N 1 1 8 "such that is in a Since a
contains infinitely many terms of ,A Ò ß , Ó Þ Ò ß , Ó ÖA ×5 # # 8"
1 1there exists in such that is in a Since a contains5 N ß 5 5 ß A
Ò ß , Ó Þ Ò ß , Ó# # " 5 # # $ $#infinitely many terms of , there
exists in such that is inÖA × 5 N ß 5 5 ß A8 $ $ # 5$Ò ß , Ó Þ ÖA
×a Continuing this process by induction, we obtain a sequence such
that$ $ 58A Ò ß , Ó 8 ÖA × ÖA ×5 5 8n is in a for each . The
sequence is a subsequence of sincen n 85 5 8 Ä A ß , Ä A + Ÿ
A Ÿ , 8n+1 n for each . Since a , and for each8 8 8 8 8
, the squeeze theorem implies that that A Ä A Þ58
