MathDB

Let

a_1, a_2, \ldots

be a strictly increasing sequence of positive integers, such that for any positive integer

n

a_n

is not representable in the for

\sum_{i=1}^{n-1}c_ia_i

for

c_i \in \{0, 1\}

. For every positive integer

m

, let

f(m)

denote the number of

a_i

that are at most

m

. Show that for any positive integers

m, k

, we have that

f(m) \leq a_k+\frac{m} {k+1}.

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