Problem. Suppose that (i) {xn} is bounded and (ii) limn→∞(xn+k−xn)=0 for every k∈N. Is {xn} convergent?
Solution.
After a moment of thought one easily deduces that a sequence converges if and only if limn→∞(xn+k−xn)=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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