Limit of difference of sequence both going to $+\infty$

Question

Let $(a_n), (b_n)$ be two sequences sucht that $a_n \rightarrow +\infty$ and $b_n \rightarrow -\infty$. Assume that for all $\epsilon > 0$ we have $$\frac{a_n}{n} \leq \epsilon$$ for $n$ large enough and for some $\xi \geq 0$$$\frac{b_n}{-n} \geq \xi$$ for $n$ large enough. How can I see that then the sequence $$C \cdot a_n + D \cdot b_n \rightarrow -\infty$$ for some constants $C,D > 0$?

Thanks a lot in advance!

Answer 1

Let $\xi >0$ and $\epsilon = \xi/2$, then, for $n$ large enough :

$$\frac {a_n + b_n}n \le \xi/2-\xi = -\frac \xi 2.$$

Then $a_n+b_n \le -n\frac \xi 2 \to -\infty$.

Answer 2

If there is $N \in \mathbb N$ such that

$\frac{a_n}{n} \leq \epsilon$ for all $ \epsilon >0$ and all $n>N$, then we have

$\frac{a_n}{n} \leq 0$ for all $n>N$, hence $a_n \leq 0$ for all $n>N$.

But then it is impossible to have $a_n \rightarrow +\infty$ !