(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let x be any real number. Prove that there exists a sequence {Rn} of rationals different from x such that {Rn} converges to x.

Use the Archimedian property, the fact that the rationals are dense in the reals, and the squeeze principle.

2. Relevant equations

The Archimedian property states that given any real number x, we may find a natural number n such that n>x.

The squeeze principle states that if the sequence {an} converges to L, the sequence {bn} converges to L, {an}<{bn} for all n, then, if {an}≤{cn}≤{bn}, then cn converges to L.

3. The attempt at a solution

Proof:

If x is rational, then let {an}=x+1/n which is a sequence of rationals that converges to x.

Assume x is irrational and without loss of generality assume x>0. Using the Archimedian property we can find a natural number n such that 0<x<n. We will now construct two sequences {an}, {bn} which approach x from below and above respectively. Take the interval [0,n] and divide it into two equal parts. Then look at the interval [a1,b1] containing x. We know that X cannot be halfway point because x is irrational. Divide this interval into two again and consider the interval [a2,b2] containing x. Continue in this manner. Take {an} to be the sequence of left endpoints:{a1,a2...} and {bn} to be the sequence of right endpoints: {b1,b2...}. Both of these sequences are monotone and both are bounded by x. Thus {an} and {bn} both converge to x. Note that {an}<{(an+bn)/2}<{bn} and by the squeeze principle, {(an+bn)/2} converges to x. #

I feel that this is the correct way to go about it but it seems to me that it is not necessary to use the squeeze principle when I've already found two sequences {an},{bn} that converge to x. However, my instructor insists that I have to use the squeeze principle so I feel as if my proof can be greatly shortened somehow and I'm just not seeing how...

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# Homework Help: Prove that There exists a sequence of rationals approaching any real number

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