Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]
We will prove by induction on p that $ |a_{pk}-p\cdot a_k|\leq p-1 $ for any $ p\geq 1$ and any k. Suppose it is true for p and prove it for p+1. But we have $ |a_{(p+1)k}-a_{pk}-a_k|\leq 1 $ and also $ |a_{pk}-p\cdot a_k|\leq p-1 $ so
$ |a_{(p+1)k}-(p+1)a_k|\leq p$ and the induction is done. Now we apply this and write $ |a_{pq}-pa_q|\leq p-1<p$ and $|a_{pq}-qa_p|\leq q-1<q $., THus, $ p+q>| pa_q-qa_p| $ and we are done.