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6
G 6
G 6
Source:
May 25, 2007
Putnam
irrational number
Irrational numbers
Problem Statement
Prove that for any irrational number
ξ
\xi
ξ
, there are infinitely many rational numbers
m
n
\frac{m}{n}
n
m
(
(
m
,
n
)
∈
Z
×
N
)
\left( (m,n) \in \mathbb{Z}\times \mathbb{N}\right)
(
(
m
,
n
)
∈
Z
×
N
)
such that
∣
ξ
−
n
m
∣
<
1
5
m
2
.
\left\vert \xi-\frac{n}{m}\right\vert < \frac{1}{\sqrt{5}m^{2}}.
ξ
−
m
n
<
5
m
2
1
.
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