# Thread: Big O, little o demonstration

1. ## Big O, little o demonstration

I must demonstrate that if $\displaystyle \{ x_n \}$, $\displaystyle \{ y_n \}$ and $\displaystyle \{ \alpha _n \}$ are sequences of real number then the following is true:
if $\displaystyle x_n=o( \alpha _n)$ then $\displaystyle x_n=O(\alpha _n )$.

My attempt: I must show that knowing that $\displaystyle \lim _{n \to \infty} \frac{x_n}{\alpha _n}=0$ then there exist a constant $\displaystyle C$ and an integer $\displaystyle r$ such that $\displaystyle |x_n|<C|\alpha _n|$ for $\displaystyle n>r$.

But to me, any constant in R that I'd take would make $\displaystyle \lim _{n \to \infty} \frac{x_n}{\alpha _n} \neq 0$. For instance if the sequence $\displaystyle \alpha_n =x_n$, then $\displaystyle \lim _{n\to \infty} \frac{x_n}{\alpha _n}=\frac{1}{C} \neq 0$. In fact C would have to be infinite for the relation to be true.

Am I missing something or the problem asks me to show a wrong result?

2. If you take $\displaystyle \alpha_n=x_n$, we don't have $\displaystyle x_n=o(\alpha_n)$ so we can't deduce the conclusion of the result is true.
The result is surely true because a convergent sequence is bounded.

3. You're right, I cannot choose my example.
To prove the afirmation, is my work valid?
What I must prove is equivalent to prove that $\displaystyle \lim _{n\to \infty} \left | \frac{x_n}{\alpha _n} \right | \leq$C. I take $\displaystyle C=0$ and I'm done?

4. Originally Posted by arbolis
You're right, I cannot choose my example.
To prove the afirmation, is my work valid?
What I must prove is equivalent to prove that $\displaystyle \lim _{n\to \infty} \left | \frac{x_n}{\alpha _n} \right | \leq$C. I take $\displaystyle C=0$ and I'm done?
This looks odd, why don't you work with the basic definition of limits: $\displaystyle |x_n|<\varepsilon |\alpha _n|$ if $\displaystyle n>N$. Now pick a c such that $\displaystyle |x_m|<c|\alpha _m|$ if $\displaystyle 0<m<N$ and just take C the maximum of both.

5. Thanks Jose. Actually I think it would be overkill.
I want to correct my previous post into this one: The existance of a constant C and an integer r such that $\displaystyle |x_n| <C|\alpha _n|$ for n>r is equivalent to write $\displaystyle \lim _{n\to \infty} \left | \frac{x_n}{\alpha _n} \right |<C$.
But I start with knowing that $\displaystyle \lim _{n\to \infty} \frac{x_n}{\alpha _n} =0$ and since $\displaystyle C \geq 0$ then the implication is obvious.

I said overkill in your base because from what I understand you want to prove it for all n while I think they ask for $\displaystyle n \to \infty$, unless I'm misunderstanding once again the exercise.

6. You're right, I missed the part where it said that only for sufficiently large n.

7. Originally Posted by Jose27
You're right, I missed the part where it said that only for sufficiently large n.
They do not state "for sufficiently large n", but this is what I believe from the definitions I have for the little and big o notations. The problem is the one I wrote in the first post in the lines before "My attempt".

By the way I didn't know the result would hold for any n, I think this is a nice result and I thank you for pointing it out.