rational and irrational in open ball

I am trying to show every open ball in R contains rational and irrational
numbers. First I suppose the center of the open ball, call it x, is
rational, then I choose positive integer n so large that x + sqrt(2)/n lies
inside the ball. But x + sqrt(2)/n is irrational so I've shown a rational
and irrational number lies in the ball.

It is the second part I'm having trouble with. Suppose the center of the
ball is irrational. I'd like to say choose n so large that x + 1/n is
inside the ball, but x + 1/n is still irrational so this doesn't help me
obtain a rational. Is there a way to "fix" the argument?



.

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