Explanation of Cauchy Proof for Convergent Sequence

[lion0001]

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

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i don't understand how they get 2e , ( a set of length 2e ?? )

 

[matticus]

what is (A + e) - (A - e)

 

[nicksauce]

It's simply (A + e) - (A - e) = 2e.
Same as I would say the interval (2,5) has length 5-2=3.

 

[lion0001]

It's simply (A + e) - (A - e) = 2e.
Same as I would say the interval (2,5) has length 5-2=3.

Excellent answer, OMG this proved that i am an idiot =(