p n / q n (where p &lt; n and q n are sequences over the naturals N ). If the limit of an (when n diverges (to infinity)) is r which is an irrational number : lim a n = r | r ∈ R ∖ Q Does that p n and q n both diverge (to infinity)?

Question

p    n        /        q    n    (where   p  &amp;lt;  n and       q    n   are sequences over the naturals       N  ). If the limit of an (when   n diverges (to infinity)) is   r which is an irrational number :  lim      a    n    =  r      |    r  ∈      R    ∖      Q  Does that       p    n   and       q    n   both diverge (to infinity)?

tabustudiofx52n · Accepted Answer

Both       P  =  {      p    n    :  n  ∈      N    }     and       Q  =  {      q    n    :  n  ∈      N    }     are bounded below by       1..If       P     is bounded above and       Q     is not bounded above, then                lim              n        →        ∞                            p        n                    q        n              =    0    ,       ,  which is not irrational.If       P     is bounded above and       Q     is bounded above, then the number of different fractions that                 p      n              q      n           can be is       ≤     max  (  P  )  max  (  Q  )  ,     which is finite (under the assumption that       P     and       Q     are both bounded above). The limit of a sequence of a finite amount of different rational numbers cannot be irrational.If       P     is not bounded above and       Q     is bounded above then                 lim              n        →        ∞                            p        n                    q        n                   is not a finite number and therefore cannot be irrational.If       P     is not bounded above and       Q     is bounded above then it is possible for                 lim              n        →        ∞                            p        n                    q        n                   to be irrational.

p n </msub> <mrow class="MJX-TeXAtom-ORD"> / </mrow> q n </msub>

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