Question

Prove that every real number is the limit of a sequence of rational numbers.

Answer 1

Attempt to prove that every real number is a limit of a sequence of rational numbers /&gt Prove that given a real number xx, there exists a rational sequence rnrn such that rn→xrn→x as nn grows.

Proof: Suppose xx is a real number. Then we know by definition, there exists a rational number such that x
Can I say x→xx→x, and x+1n→xx+1n→x as nn grows. Thus by the sandwich theorem, rn→xrn→x?

Or should I start with, let ε>0ε>0. Then we need to show |rn−x|<ε|rn−x|<ε?