Every convergent sequence is a cauchy sequence(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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the part where i have trouble understanding this proof is , where does the e/2 comes from?

in other words how does e/2 appears in the proof,

if the answer is this " the difference between a_n , a_m was less than twice the original choice of e " i would like to see the arithmetic that produces e/2

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# Cauchy sequence

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