# Infinite Finite Difference

Calculus Level 4

Let $$a_n$$ be a real sequence with positive terms such that $$\displaystyle \lim_{n\to \infty} a_n = 0$$.

If the sequence $$c_n = a_n - a_{n+1}$$ is nonincreasing, what is the strongest statement we can make about the limit $$\displaystyle \lim_{n\to \infty} n c_n$$ (that is true for any choice of $$a_n$$)?

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