Thread: Neighbourhood of an irrational

Could someone point me towards any proof for whether or not the neighbourhood around an irrational number is rational? Would the rational numbers being a dense subset of the real numbers cause there to be a rational arbitrarily close to any irrational?

Thanks!

Originally Posted by svanbeek

Could someone point me towards any proof for whether or not the neighborhood around an irrational number is rational? Would the rational numbers being a dense subset of the real numbers cause there to be a rational arbitrarily close to any irrational?

Neighborhoods are neither rational nor irrational.
Do you mean that any neighborhood of an irrational contains a rational?
The answer to that is yes. The rationals are dense in the reals.