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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §2.6 Limits, #39 · Calculus · Limits

No matter what decimals come later, … approaches a limit

No matter what decimals come later, $a_{1} = .4, a_{2} = .49, a_{3} = .493, \dots$ approaches a limit $L$ . How do we know (when we can't know $L$ )? Cauchy's test is passed: the $a$ 's get closer to each other.
(a) From $a_{4}$ onwards we have $∣ a_{n} - a_{m} ∣ <$ ____.
(b) After which $a_{N}$ is $∣ a_{m} - a_{n} ∣ < 1 0^{- 7}$ ?

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §2.6 Limits, #39

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