[M@gpie]

ผมลองทำดูนะครับไม่รู้ว่าถูกไหม ผิดยังไงก็บอกนะครับผม



We will prove the contrapositive of this statement instead.
Assume that $\lim_{x\rightarrow \infty} \frac{f(x)}{x} \neq a$.
Then there exists $\epsilon >0 $ for any $M >0$ such that if $x > M$ and $\left| \frac{f(x)}{x} - a \right| \geq \epsilon$.
Hence, we can see that there exists $\epsilon >0$, for any $M > 0$ such that for any $n\in \mathbb{N}$, if $n > M$ and $\left| \frac{f(n)}{n} - a \right| \geq \epsilon$. This implies $\lim_{n\rightarrow \infty} a_n \neq a.$ The proof is finished.