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Can we find this particular infinite sequence?

Source: IMO LongList 1988, Vietnam 2, Problem 90 of ILL

November 9, 2005
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Problem Statement

Does there exist a number α,0<α<1\alpha, 0 < \alpha < 1 such that there is an infinite sequence {an}\{a_n\} of positive numbers satisfying 1+an+1≤an+αn⋅αn,n=1,2,…? 1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots?
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