KURSCHAK 1960 Problem 2

Let $a_1 = 1, a_2, a_3,...$ : be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$ . Prove that every positive integer can be expressed as a sum of $a_i$ s.