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Studying for the final and there are two questions from the beginning of the semester that I can't seem to find in my notes/book:

1) Give an example of a divergent sequence $(a_n)$ such that $\lim_{n \to \infty} (a_{n+1} - a_n) = 0$. Why is your example not a Cauchy sequence?

2) If A and B are non-empty sets, give the definition of A x B.

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1) Let $a_n = H_n$ where $H_n$ is the sum of the harmonic series up to the $n$th term. Clearly $a_{n + 1} - a_n = 1/(n+ 1) \to 0$ as $n \to \infty$. The harmonic series are not convergent, and thus not Cauchy.

2) $A \times B = \{(a, b) | a \in A \land b \in B\}$

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    $\begingroup$ Or just let $a_n=\ln(n)$. $\endgroup$
    – bof
    May 4, 2014 at 2:04
  • $\begingroup$ @bof, but with tests, since it's more difficult to do bound $\ln(n)-\ln(n+1)<\epsilon$, it's better to Keep It Simple Stupid. $\endgroup$
    – user18862
    May 4, 2014 at 4:09
  • $\begingroup$ @NeuroFuzzy $\ln(n+1)-\ln(n)=\ln(\frac{n+1}n)=\ln(1+\frac1n)\to\ln(1)=0$. $\endgroup$
    – bof
    May 4, 2014 at 4:52
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  1. Hint: think about series. I can give more of a hint if you like.

  2. $A \times B$ is defined as the set of ordered pairs $(a, b)$, where $a \in A$ and $b \in B$. See here: Cartesian product

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