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.

Question

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.

Brennen Bishop · Accepted Answer

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&#039;ll get the same contradictions with             q      n              p      n        →      1    a  .

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

Answered question

Answer & Explanation

New Questions in Algebra I