On Words Containing all Short Subwords

Lutz Priese raised the following conjecture: Almost all words of length n over a finite alphabet A with m letters contain as subwords all words of length ⌊log log n⌋ over A as n → ∞. In this note we prove that this property holds for subwords of length k(n) over A provided limn → ∞ k(n)log n = 0.