Suppose (an) is Cauchy and that for all m, k ≥ 100, |am − ak| < 1/4. Let E > 0. Does it follow that the interval (a100 − E, a100 + E) contains infinitely many terms of (an)? Either show that it does or give a counter-example.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
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3. Suppose (an) is Cauchy and that for all m, k ≥ 100, |am − ak| < 1/4. Let E > 0. Does it follow that the interval (a100 − E, a100 + E) contains infinitely many terms of (an)?
Either show that it does or give a counter-example.

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