Let a ∈ ( 0 , 1 ) be an irrational number and suppose there exists a

Chaz Blair

Chaz Blair

Answered question

2022-05-19

Let a ( 0 , 1 ) be an irrational number and suppose there exists a sequence
r n = p n q n with p n , q n N and lim n r n = a.
Show that both { p n : n N } and { q n : n N } are not bounded.
Suppose a Q , a ( 0 , 1 ) , and sequence r n = p n q n a, with p n , q n N . Show that both { p n : n N } , { q n : n N } are not bounded.

Answer & Explanation

Brennen Bishop

Brennen Bishop

Beginner2022-05-20Added 6 answers

Suppose { p n } is bounded.
If { q n } is also bounded, then the set { p n q m n , m N } Q is finite, thus it cannot have an irrational limit point.
So { q n } must be unbounded. But then, there exists a subsequence q n k . Thus p n k q n k 0, which contradicts a ( 0 , 1 ).
If you suppose first that { q n } is bounded, you'll get the same contradictions with q n p n 1 a .

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