Say, $c$ with $c>0$, is an irrational number, what I need to prove is, any $\delta$-ngbhd of $c$ i.e. $V_{\delta}(c)$ contains a finite number of rational numbers.
I am not sure if the statement is correct or not.
Could anyone comment/provide a hint for the proof ?
If you have $5$ cars, do you have $3$ cars?
By denseness, we know that the neighborhood indeed contains infinitely many rational numbers.
Does the neighborhood contains $5$ rational numbers?
I would view "the neighborhood containly only finitely many numbers" as a False statement.
But we can certainly find finitely many numbers in that neighborhood.