×

# Difference of Indices in a Sequence

A sequence $$x_1, x_2, x_3, \dots$$ has the following properties:

(a) $$1=x_1 < x_2 < x_3 \dots$$;

(b) $$x_{n+1} \leq 2n$$ for all $$n \in \mathbb{N}$$.

Prove that for each positive integer $$k$$ there exist indices $$i$$ and $$j$$ such that $$k = x_i −x_j$$.

Note by Finn Hulse
2 years, 5 months ago