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Equivalent sequnces: boundness and Cauchy

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5.2.1. Show that if (a_n)[n=1,infinity] and (b_n)[n=1,infinity] are equivalent sequences of rationals, then (a_n)[n=1,infinity] is a Cauchy sequence if and only if (b_n)[n=1,infinity] is a Cauchy sequence.

5.2.2. Let epsilon > 0. Show that if (a_n)[n=1,infinity] and (b_n)[n=1,infinity] are eventually epsilon-close, then (a_n)[n=1,infinity] is bounded if and only if (b_n)[n=1,infinity] is bounded.

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Solution Summary

We prove that
1) If two sequences are euivalent, then if one of then is Cauchy, then so is the other one.
2) If two sequences are eventually epsilon-close, then one of them is bounded if and only if the other one is.
The solutions are in a PDF file.