If $|f(p+q)-f(q)|\leq p/q$ for rational $p$ and $q$ (with $q\neq 0$), then $\sum_{i=0}^k |f(2^k) -f(2^i)| \leq k(k-1)/2$

Question

If $\left|f(p+q) - f(q)\right|\leq p/q$ for all rational $p$ and $q$, with $q \neq 0$, then prove that $$\sum_{i=0}^k \left|f\left(2^k\right) -f\left(2^i\right)\right| \leq \frac12 k(k-1)$$

My try:

I consider the sum for $i=r $ which gives the inequality from given property of function $$ \left |f\left(2^k\right) -f\left(2^i\right)\right| \leq 2^{k-i} - 1$$, and then summed it, but it doesn't work.

Answer 1

By triangle inequality

$|f(2^k)-f(2^j)| \leq |f(2^k)-f(2^{k-1})|+...|f(2^{j+1})-f(2^j)|\leq k-j$

Hence $\sum_{i=0}^{i=k}{|f(2^k)-f(2^i)|} \leq \sum_{i=0}^{i=k}{k-i} \leq 0.5(k)(k+1)$

Still not as good as $0.5(k)(k-1)$