How to prove that it is a Cauchy sequence?

Question

Let $(a_n)\in\mathbb{R}^{\mathbb{N}}$ such that : $\forall (p,q)\in \mathbb{N}^2$, $\vert a_{p+q}-a_p -a_q \vert \le 1$.

Show that $(\frac{a_n}{n})_{n\in \mathbb{N}^*}$ is a Cauchy sequence (without using subadditivity lemma).

Here's my attempt : we can notice that in particular the real sequence $(b_p=a_{p+1}-a_p)$ is bounded. We can extract a convergent subsequence $(b_{\varphi(p)}=a_{\varphi(p)+1}-a_{\varphi(p)})$. Suppose that $b_{\varphi(p)}\rightarrow l\in \mathbb{R}$.

Answer 1

The topic is related to the Hyers-Ulam-stability of the Cauchy functional equation. See my answer to A sequence satisfying $\lvert f_{i+j}-f_i-f_j\rvert \leq M$ bounded?.