- #1

Lelouch

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## Homework Statement

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The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128).

##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real numbers. Then, it is not possible for ##(a_n)^\infty_{n=m}## to converge to ##l## while also converging to ##l'##.

## Homework Equations

/definitions/propositions[/B]##Defintion.## Let ##x, y## be real numbers. The distance ##d(x, y)## between ##x## and ##y## is defined by $$d(x, y) := |x - y|.$$

##Proposition##. Let ##x, y, z## be real numbers. We have

(a) $$d(x, y) = d(y, x),$$

(b) $$d(x, z) \leq d(x, y) + d(y, z).$$

##Defintion.## Let ##l## be a real number. A sequence ##(a_n)^\infty_{n=m}## of real numbers

**converges to ##l##,**##\lim_{n \rightarrow \infty} {a_n} = l##, iff. for every real ##\epsilon > 0## there is an ##N \geq m## such that for all ##n##

$$n \geq N \Rightarrow d(a_n, l) \leq \epsilon.$$

## The Attempt at a Solution

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##Proof.## Suppose for the sake of a contradiction that ##\lim_{n \rightarrow \infty} {a_n} = l## and that ##\lim_{n \rightarrow \infty} {a_n} = l'##. We then have,

$$(\forall \epsilon > 0)(\exists N \geq m)(\forall n)(n \geq N \Rightarrow d(a_n, l) \leq \epsilon)$$

and

$$(\forall \epsilon > 0)(\exists N' \geq m)(\forall n)(n \geq N' \Rightarrow d(a_n, l') \leq \epsilon).$$

If we let ## n' := max(N, N')##, then we have

$$(\forall \epsilon > 0)(\exists n' \geq m)(\forall n)(n \geq n' \Rightarrow d(a_n, l) \leq \epsilon \land d(a_n, l') \leq \epsilon).$$

Thus, by the triangle inequality and symmetry of distance we have

$$ d(l, l') \leq d(a_n, l) + d(a_n, l') \leq \epsilon + \epsilon = 2 \epsilon.$$

Hence, ## d(l, l') \leq 2 \epsilon.## Since ## l \neq l' ##, we have ##d(l, l') > 0##. If we choose ##\epsilon = \frac {d(l, l')} 3##, we then arrive at ##d(l, l') \leq \frac {2d(l, l')} 3##, a contradiction. Therefore, it is not possible to converge to both ##l## and ##l'##.My questions are.

(i) First and foremost. Is the proof of the proposition correct.

(ii) I am a little irritated about the "If we let ## n' := max(N, N')##" part. Why is one allowed to do this? And how does one come up with this idea of taking the ##max## of two numbers in order to proceed throughout the proof. I have seen also the ##min## being used in proofs for limits of functions.

(iii) I see that in the proof supplied by Terence Tao, he adds the "If we choose ##\epsilon = \frac {d(l, l')} 3##" part at the beginning of the proof. Why so? Surely, he did not know this at the beginning of the proof did he? I only saw this "opportunity" once I was at the end of the proof. I have seen such ""covering the tracks" procedures in many proofs. Why do we do this? Is it not allowed to leave the proof the way I have?