Answer to Question #161764 in Calculus for Vishal

Question

Let {an}∞n=1 be a convergent sequence. Prove that {an}∞n=1 satisfies Cauchy’s criterion

Answer 1

$\mathrm{Given:\;\{a_n\}_{n=1}^{\infty}\;is\;a\;convergent\;sequence.}$

$\mathrm{Prove:\{a_n\}_{n=1}^{\infty}\;satisfies\;Cauchy's\;criterion.}$

$\mathrm{Proof:\;Let\;\epsilon >0\;be \;given.Since\;\{a_n\}_{n=1}^{\infty}\;is\;a\;convergent\;sequence.\;So,}$

$\mathrm{choose\;N\;such\;that\;if\;n>N,\;we\;have\;|a_n-\alpha |<\dfrac{\epsilon}{2}\;.}$

$\mathrm{Then\;if\;m,n>N,we\;have\;|a_n-a_m|=|a_n-\alpha+\alpha-a_m|}$

$\mathrm{\leq |a_n-\alpha|+|a_m-\alpha|}$

$\mathrm{< \dfrac{\epsilon}{2}+\dfrac{\epsilon}{2}}$

$\mathrm{=\epsilon}$

$\mathrm{\therefore \{a_n\}_{n=1}^{\infty}\;is\;a\;cauchy\;sequence.}$

$\mathrm{}$