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I stumbled upon the notion of "fundamental sequence" and a theorem (Cauchy I guess) that says that a fundamental sequence is convergent. Right?

Can anybody tell me how to prove the fundamental-ness of a sequence? Thanks

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A Cauchy sequence in a complete space converges. $(x_n)$ is Cauchy if, for every $\epsilon>0$, there is an $M$ such that $d(x_n,x_m)<\epsilon$ for all $n,m\ge M$. –  David Mitra Nov 19 '11 at 8:35
    
@DavidMitra for me this actually means that the bigger the n and m the smaller the difference of the two terms. Right? –  Andrew Nov 19 '11 at 8:40
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Yes, that's right. The terms of the sequence get closer and closer to each other as $n$ and $m$ get large. –  David Mitra Nov 19 '11 at 8:42
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The sequence $(1/n)$ converges to $0$. The series $\sum (1/n)$ does not converge, meaning that the sequence of partial sums does not converge. –  André Nicolas Nov 19 '11 at 8:50
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@andrew $(1/n)$ converges to 0... As for Cauchy implies convergent, it depends on what the underlying space is. If the space is $X=(0,1]$, then $(1/n)$ is Cauchy but not convergent. If the space is $X=[0,1]$, then every Cauchy sequence is convergent. Spaces for which the preceeding property holds are called $complete$. –  David Mitra Nov 19 '11 at 8:51

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