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Saturday, October 18, 2014

A Problem

Problem. Suppose that (i) {xn} is bounded and (ii) limn(xn+kxn)=0 for every kN. Is {xn} convergent?

Solution.
After a moment of thought one easily deduces that a sequence converges if and only if limn(xn+kxn)=0 uniformly in k. Therefore if every such sequence converges, the ``pointwise'' convergence in k and ``uniform pointwise" convergence in k will be the same, this, by math instinct, is difficult to be true.

So we raise a nonconvergent example as follows. Define xn:=nπ0sinxdx, this is easily seen that {xn} satisifies the two conditions above, but {xn} diverges.

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