Is it true that for any sequence of irrational numbers { h n } converges to zero in [ − h , h ] both α + h n and α − h n and α + h n and α − h n are always irrationals ?

Question

Is it true that for any sequence of irrational numbers   {      h    n    } converges to zero in   [  −  h  ,  h  ]   both     α  +      h    n          and      α  −      h    n     and     α  +      h    n          and      α  −      h    n     are always irrationals ?

SweallySnicles3 · Accepted Answer

Most certainly not.In fact, take any sequence       q    n   of rational numbers in   [  α  −  h  ,  α  +  h  ] that converges to   α   [  1  ]. Now, define       h    n    =      q    n    −  α. Then,      h    n   is in   [  −  h  ,  h  ].      h    n   converges to   0.for all   n  ∈      N  , hn is irrationalHowever,   α  +      h    n    =      q    n  , which means that   α  +      h    n   is always rational.

Is it true that for any sequence of irrational numbers <mo fence="false" stretchy="false">{

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