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

+

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

+