IE 500 – Fall 2009
E. Alper Yıldırım
HOMEWORK 4 (due Wednesday, November 18 in class)
1. Give a complete characterization of Cauchy sequences of
integers
. Discuss what kind of real numbers
can be represented by such Cauchy sequences.
2. Let (
x
n
) be a Cauchy sequence of real numbers.
Consider the seqence (
y
n
) given by
y
1
= 0 and
y
n
=
x
n

x
n

1
,
n
= 2
,
3
, . . .
.
(a) Prove that (
y
n
) is a Cauchy sequence of real numbers.
(b) Find the limit of the sequence (
y
n
) and justify your result.
3.
(a) Let
a
and
b
be any two real numbers. Prove that

a
  
b
 ≤ 
a

b

.
(b) Let (
x
n
) be a Cauchy sequence of real numbers. Using part (a), prove that (
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 Spring '10
 zeynephuygur
 Real Numbers, yn, Limit of a sequence, Cauchy sequence

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