qn
1.1.15. Let a be irrational. Show that A = {m + not : m, n G Z}
is dense in R, i.e. in any open interval there is at least one element
of A.
1.1.16. Show that {cosn : n G N} is dense in [—1,1].
1.1.17. Let x G R \ Z. Define the sequence {xn} by setting
x = [x] + , xi = [xi] + ,..., x
n
_i = [xn_i] -h —.
X\ X2 Xn
Then
X =
\X
+
N +
[x2] +
•• -f
[xn_i] -h
Show that x is rational if and only if there exists n G N for which
xn is an integer.
Remark. The above representation of x is said to be a finite con-
tinued fraction. The expression
a0 +
a\
+
a2 +
1
" +
Gn-1
+
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