Suppose that a sequence of (rational) fractions p / q converge to an irrational number r. Show that q converges to infinity.

Question

Suppose that a sequence of (rational) fractions   p      /    q converge to an irrational number   r. Show that   q converges to infinity.

mallol3i · Accepted Answer

Let       r    n    =      p    n        /        q    n  .       r    n    →  x.   x  ∉      Q  .Suppose       q    n    →  y. Then for   1      /    2  &amp;gt;  ϵ  &amp;gt;  0 there exists and   N such that      |        q    n    −      q    m        |    ≤      |        q    n    −  y      |    +      |    y  −      q    m        |    &amp;lt;  ϵ  +  ϵ  &amp;lt;  1 for all   m  ,  n  &amp;gt;  N. But       q    n    ,      q    m   are integers so       |        q    n    −      q    m        |    &amp;lt;  1    ⟺        q    n    =      q    m  . So there exists an   N where all       q          n      ;      n      &amp;gt;      N       are an equal constant integer,   y.If       r    n    →  x then   {      r    n    ∗  y  }  →  x  y. For   n  &amp;gt;  N,       r    n    ∗  y  =      p    n    ∗  y      /        q    n    =      p    n    ∗  y      /    y  =      p    n    ∈      Z  . So, by the exact same argument as above:       r    n    ∗  y becomes a sequence of integers for   n  &amp;gt;  N. As this converges there is an   M where       r    n    ∗  y is an equal constant integer,   p, for all   n  &amp;gt;  M.So for   n  &amp;gt;  M, r      r    n    ∗  y  =  p    ⟹        r    n    =  p      /    y and       r    n    →  p      /    y  =  x  ∈      Q  .This A contradiction.

Lydia Carey · Accepted Answer

Let   {  (      p    q        )    k    } be a sequence of of rational numbers such that   {  (      p    q        )    k    }  =            {              p        n            }              {              q        n            }       and so that the sequence converges to   r  ∈      R    −      Q  .Suppose that       lim          n      →      ∞            q    n    =  Q.If       p    n   converges, we have a contradiction to the irrationality or   r, since this would imply that it converges to some   z  ∈      Z  .If       p    n   diverges,             p      n        Q   also diverges to infinity, contradicting the fact that the sequence is convergent.

Suppose that a sequence of (rational) fractions p

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