(Easy to see this weighted mean is constant, and thus only possible limit. "better way?" is re: demonstrating convergence)
Let x_∞,...,x_(∞+N) be the limit of a convergent subsequence of adjacent (N+1)-tuples of elements of the sequence.
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Variance{i copies of x_(∞+i-1) for i=1,...,N} = Variance{i copies of x_(∞+i) for i=1,...,N}, but would be > if not 0.
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In "decreases as i increases", "i" should be "k".
End of conversation
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