calculus - How to show $limlimits_{ntoinfty} frac{a_n}{n}=2 Rightarrow limlimits_{ntoinfty}(a_n-n)=infty$?

I find it hard to answer the question below. I just don't know how to use the fact that $$\lim_{n\to\infty} \frac{a_n}{n}=2.$$ Maybe with limit arithmetic?

Let $(a_n)$ be a sequence, where $$\lim_{n\to\infty} \frac{a_n}{n}=2$$

Is it correct that $$\lim_{n\to\infty}(a_n-n)=\infty$$

I think it is correct since from limit arithmetic I can get to the conclusion that $$\lim_{n\to\infty}a_n=2\lim_{n\to\infty}n$$

But I just can't prove it.

Thanks.

Answer

Hint: Prove that $\dfrac{a_n}{n} > 1.5$ for all $n$ sufficiently large.

Solution:

Since $\dfrac{a_n}{n} \to 2$, taking $\varepsilon=0.5$, we get that $\dfrac{a_n}{n} > 1.5$ for all $n$ sufficiently large. This implies that $a_n-n> 0.5 n$ for all $n$ sufficiently large and so $a_n -n \to \infty$.