Problem of the Week

Let Y_k be the number of positive integers less than 10^k at least one of whose digits is a 9, and N_k be the number of positive integers less than 10^k none of whose digits is a 9.  For example, (N₁,Y₁) = (8,1),  (N₂,Y₂) = (80,19),  (N₃,Y₃) = (728,271).  From this evidence it appears as though Y_k £ N_k. What is the limit of Y_k/N_k , as k goes to infinity?

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