In the book "The arithmetic of elliptic curves" by Silverman, I try to solve the exercise 6.2.:
Let $\Lambda=\{n\omega_1+m\omega_2:m,n\in\mathbb{Z}\}$ a lattice of $\mathbb{C}$. Then exists a costant c such that for all $n\in\mathbb{N}^+$ $\lvert\{ω∈Λ:n⩽|ω|<n+1\}\rvert\le cn$.
My idea is, said $\delta$ the minimum distance between 2 distinct point of $\Lambda$, then the set $A_r=\{\omega\in\Lambda :r⩽ \lvert \omega\rvert<\frac{\delta}{2} \}$ contains at most $\frac{4\pi r}{\delta}$ point (because any two distinct lattice point of $A_r$ must be separated by an arch of length at least $\frac{\delta}{2}$). Let $N>\frac{2}{\delta}$. Then $\{ω∈Λ:n⩽|ω|<n+1\}\subseteq\bigcup\limits_{k=0}^NA_{n+k\frac{\delta}{2}}$ but $\sum\limits_{k=0}^N \lvert A_{n+k\frac{\delta}{2}}\rvert\le \sum\limits_{k=0}^N\frac{4\pi}{\delta}(n+k\frac{\delta}{2})=\frac{4\pi N}{\delta}n+\frac{N(N+1)}{2}\frac{\delta}{2}$.
Now the second addend is inferior to $n$ only $n$ is sufficiently big. It's this reasoning correct? and how prove the assert for every $n$?