Why Is e/2 Used in the Cauchy Sequence Proof?

[lion0001]

This is what it is in my book, i don't know how it went from saying that the difference is 2e and then using e/2 . The part that confuses me , is why the book outlines (1) to then use it in the proof, it looks that (1) doesn't even matter because since e > 0 then e/2 will do just fine.

(1) Suppose { a_n } converges to A. Choose e > 0, THere is a positive integer N such that, if
n, m >= N , then A - e < a_n < A + e and A - e < a_m < A + e
Thus for all n, m >= N we find a_n ∈ ( A - e , A + e ) and
a_m ∈ ( A - e , A +e ) . the set ( A - e, A +e ) is an interval of length 2e ,
hence the difference between a_n, and a_m is less then 2e

we will now state a theorem , the proof of which we have just outlined


THEOREM :Every convergent sequence is a cauchy sequence

Proof: Suppose {a_n} converges to A, choose e > 0 , then e/2 > 0

there is a positive integer N such that n>=N implies abs[a_n - A ] < e/2

by (1) the difference between a_n , a_m was less than twice the original choice of e

Now if m , n >= N , then abs[a_n - A ] < e/2 and abs[a_m - A] < e/2 , hence

abs[a_n - a_m ] = abs[ a_n - A + A - a_m ] <= abs[a_n - A ] + abs[ A - a_m ]
= abs[a_n - A ] + abs[ a_m - A ] < e/2 + e/2 = e
thus {a_n} is cauchy

 

[mutton]

You want your conclusion to be |a_n - a_m| < e to satisfy the definition of a Cauchy sequence. It is not always easy to get e by itself in the inequality right away. (1) is rough work, after which you divide all expressions containing e by 2 to get the proof.

If you understand the proof, then don't worry about (1) so much. Looking at other epsilon proofs will help.

 

[lion0001]

You want your conclusion to be |a_n - a_m| < e to satisfy the definition of a Cauchy sequence. It is not always easy to get e by itself in the inequality right away. (1) is rough work, after which you divide all expressions containing e by 2 to get the proof.

If you understand the proof, then don't worry about (1) so much. Looking at other epsilon proofs will help.

I think i am going to follow your advise Mutton, (1) seems not to be connected to the proof.
In the proof if e > 0 then it must be true for e / 2 , and e/2 is used because at the end the answer is e

Thanks again Mutton.