Can Rational Numbers Approximate Irrational Numbers Arbitrarily Closely?

[Kartik.]

Prove the theorem comprising that an irrational number β can be described to any limit of accuracy with the help of rational.

Attempt-
Taking the β to be greater than zero and is expressed with an accuracy of 1/n
For any arbitrary value of β, it falls between two consecutive integers which are assumed to be N and N+1.Division of the interval between N and N+1 is into n parts.After this step i know that the number will fall between N+m/n and N+(M+1/n), but my question is HOW? and i also want to know about the rest of the proof.

 

[DonAntonio]

Prove the theorem comprising that an irrational number β can be described to any limit of accuracy with the help of rational.

Attempt-
Taking the β to be greater than zero and is expressed with an accuracy of 1/n
For any arbitrary value of β, it falls between two consecutive integers which are assumed to be N and N+1.Division of the interval between N and N+1 is into n parts.After this step i know that the number will fall between N+m/n and N+(M+1/n), but my question is HOW? and i also want to know about the rest of the proof.

Please define what is "a number can be described with the help of a rational".

I bet, for sure, that what you need here is a simple fact about limits, the euclidean topology of the reals and stuff, but if

you haven't yet studied this then it's important to know what you think you have to prove.

DonAntonio

 

[HallsofIvy]

I assume you mean that, given an irrational number, x, and \delta> 0, there exist a rational number, y, such that |x- y|< \delta.

How you would prove that depends upon how you are defining "real number". If, for example, you define the real numbers as "equivalence classes of Cauchy sequences of rational numbers" this is relatively simple to prove.

 

[mathman]

If you start with the decimal expansion of an irrational number, you can truncate it at any time to get a rational approximation. At n decimal places the error term ~ 10^-n.

 

[Diffy]

Edit: Posted in wrong thread. Sorry.

 

[laughingebony]

For any arbitrary value of β, it falls between two consecutive integers which are assumed to be N and N+1.

Can you prove this? If you can, see if you can use the same technique to prove that β falls between N+(m/n) and N+[(m+1)/n], for some m.