Is the space $c_0$ of all infinitesimal numeric sequences $a = (a_1, a_2,...,a_n,...)$ complete? The norm is $||a|| = sup|a_i|$.

As i couldn't prove that every Cauchy sequence has a limit in $c_0$ (so it'd be complete), I've been trying to find such Cauchy sequence that does not have a limit in $c_0$ to prove its incompleteness. But I haven't succeeded.

  • $\begingroup$ This may be a stupid question but, what is a numeric sequence? $\endgroup$ – Don Thousand Apr 6 at 16:49
  • $\begingroup$ @Don Thousand , I meant just a usual sequence of numbers (not functional or other) $\endgroup$ – SilverLight Apr 6 at 17:34
  • $\begingroup$ But, by definition of infinitesimal sequence, every sequence converges to 0? $\endgroup$ – Don Thousand Apr 6 at 17:45
  • $\begingroup$ @ Don Thousand , sure $\endgroup$ – SilverLight Apr 6 at 17:51

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