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## Main Question or Discussion Point

As far as I understand, a sequence converges if and only if it is Cauchy. So say for some sequence a_n and for all epsilon greater than zero we have [tex] |a_n - a_{n+1}| < \epsilon [/tex] for large enough n.

We could then say a_n converges if and only if [tex] \lim_{n \rightarrow \infty} a_n - a_{n+1} = 0 [/tex].

But what about if a_n = ln(n)?

[tex] ln(n) - ln(n+1) = ln(n/(n+1)) [/tex] so for n tending to infinity ln(n) - ln(n+1) goes to 0. So I should be able to say that the sequence converges, but ln(n) obviously goes to infinity for increasing n.

What's the mistake in my reasoning?

We could then say a_n converges if and only if [tex] \lim_{n \rightarrow \infty} a_n - a_{n+1} = 0 [/tex].

But what about if a_n = ln(n)?

[tex] ln(n) - ln(n+1) = ln(n/(n+1)) [/tex] so for n tending to infinity ln(n) - ln(n+1) goes to 0. So I should be able to say that the sequence converges, but ln(n) obviously goes to infinity for increasing n.

What's the mistake in my reasoning?