Let $f$ be real valued function on $\Bbb R$. Let $g(x,y)＝f(x＋y)-f(x)-f(y)$. Supposed that g is bounded on $\Bbb R^2$

Question

Let $f$ be real valued function on $\Bbb R$. Let $g(x,y)＝f(x＋y)-f(x)-f(y)$・・・①. Supposed that g is bounded on $\Bbb R^2$.

Let fix $a∈\Bbb R$, $a_n＝1/2^n f(2^na)$

I would like to prove $｛a_n｝$ is cauchy sequence.

I tried to prove $a_m-a_n →0$,when $m＞n→∞$, but I don't know the timing to use ①.

Thank you in advance.

Answer 1

Let say $g$ is bounded by $C$. Then you have $$|f(2x)-2f(x)| \le C$$ for all $x \in \mathbb R$. This gives you $$|a_{n+1}-a_n| = \frac{|f(2\cdot 2^n a)-2f(2^n a)|}{2^{n+1}} \le \frac{C}{2^{n+1}}.$$ Now work out what that means for $|a_n-a_m|$ and you are done.