# Prove convergence using the limit definition

Suppose that $a_n \rightarrow \infty$ and $b_n \rightarrow L$ as $n \rightarrow \infty$, where $L$ is real. Prove from the limit definitions that

(a) $b_n/a_n \rightarrow 0$ as $n \rightarrow \infty$

(b) $a_n+b_n \rightarrow \infty$ as $n \rightarrow \infty$

Sorry guys. This is fairly easy. Just can't figure it out.

I know how to rigorously define each of the separate components, but I am unsure of how to put it together.

(a) Consider the case where $L \neq 0$. If n is sufficiently large, then for any $\epsilon > 0$

$$||b_n| - |L|| \leq |b_n - L| < |L|/2 \implies |b_n| < 3|L|/2,$$

and

$$a_n > \frac{3|L|}{2\epsilon}.$$

Hence,

$$\left|\frac{b_n}{a_n}\right| < \epsilon.$$

Make a similar argument for $L = 0$. For any $\epsilon > 0$, if $n$ is sufficiently large we have $|b_n| < \epsilon$ and $a_n > 1$. Whence, $|b_n/a_n| < \epsilon$.