# Understanding and writing limit proofs

I got this question :

Consider $a_n,\:b_{n\:}$ sequences such that for every n , $0\le a_n\le \:b_{n\:}$.

Let $\lim _{n\to \infty }\left(\frac{b_n}{a_n}\right)\:=\:1$, and $a_n$ is a bounded sequence.

Prove that $\left(a_n-b_n\right)_{n=1}^{\infty \:}\:\rightarrow \:0$.

I tried little bit by myself to understand what is given:

By definition of limit, we see that : $\forall\epsilon, \exists n_{0} \forall n> n_{0}\Rightarrow |\frac{b_{n}}{a_{n}}- 1|< \varepsilon$, and also that exist some $M$ that for every $n$, $-M<a_n<M$.

Now i'm looking for that right?: i need to prove that $\forall\epsilon, \exists n_{0} \forall n> n_{0}\Rightarrow |a_{n}-b_{n}- 0|< \varepsilon$

Here is where i struggle: i choose some $\epsilon$. What i need to find? an $N2$ that for every $n>N2, |(a_{n}-b_{n})- 0|< \varepsilon$ ?

How to start?

Hint: use may the fact that $|a_n-b_n|=|a_n||\frac{b_n}{a_n}-1|$ in your proof .

• Yes, but what i actually need to write to do it formal? – user2637293 Sep 18 '14 at 1:09
• I would recommend you to fill in the gap yourself. but if not interested you can look at the proof provided by @Ishfaaq down here. – BigM Sep 18 '14 at 1:12

Hints

First of all notice that $\left|{\dfrac{b_n}{a_n} - 1}\right| = \dfrac{|(b_n - a_n) - 0|}{|a_n|}$

Secondly note that $|a_n| \le M \;\; \forall n \in \Bbb N$

Now let $\epsilon \gt 0$ be arbitrary and think what you can do with a quantity like $$\dfrac{\epsilon}{M} \gt 0$$

UPDATE: Just extract what the definition of $\lim \dfrac{a_n}{b_n}$ means. It says for every $\epsilon \gt 0$ there is $n_0$ such that the difference between $\dfrac{a_n}{b_n}$ and the limit ($= 1$ ) is less than $\epsilon$. So all you need to write in your paper would be to let $\epsilon$ be arbitrary. Then since $\lim \dfrac{a_n}{b_n}$ is given, choose $\dfrac{\epsilon}{M}$ as your arbitrary positive quantity. Then as per the definition of the limit there exists $n_0 \in \Bbb N$ such that $$n \ge n_0 \implies \left|{\dfrac{b_n}{a_n} - 1}\right| = \dfrac{|(b_n - a_n) - 0|}{|a_n|} \lt \dfrac{\epsilon}{M} \implies | (b_n - a_n ) - 0 | \lt |a_n | \cdot \dfrac{\epsilon}{M} \le \epsilon$$

And you're done!

• This is the problem. i don't know what to write after i choose arbitrary epsilon. i need to someone write the answer like in a test. it's not the question of HOW to do this, its about WHAT to write to do it formally, tnx! – user2637293 Sep 18 '14 at 1:15
• @user2637293: I edited the answer. Comment if you need more help. – Ishfaaq Sep 18 '14 at 6:58