1. ## Sequence Question

Prove the following:

If $\displaystyle \{b_n\}_{n=1}^{\infty}$ converges to $\displaystyle B \neq 0$ and $\displaystyle b_n \neq 0$ for all $\displaystyle n$, then there is an $\displaystyle M > 0$ such that $\displaystyle |b_n| \geq M$ for all $\displaystyle n$.

Now, this sounds very similar to a Lemma that was proven in class.

If $\displaystyle \{b_n\}_{n=1}^{\infty}$ converges to $\displaystyle B$ and $\displaystyle B \neq 0$, then there is a positive real number $\displaystyle M$ and a positive integer $\displaystyle N$ such that, if $\displaystyle n \geq N$, then $\displaystyle |b_n| \geq M$.

I'm just having trouble generalizing the above for all n... Does it have something to do with saying that $\displaystyle b_n \neq 0$?

2. ## Re: Sequence Question

Originally Posted by FishStyx
Prove the following:

If $\displaystyle \{b_n\}_{n=1}^{\infty}$ converges to $\displaystyle B \neq 0$ and $\displaystyle b_n \neq 0$ for all $\displaystyle n$, then there is an $\displaystyle M > 0$ such that $\displaystyle |b_n| \geq M$ for all $\displaystyle n$.

Now, this sounds very similar to a Lemma that was proven in class.

If $\displaystyle \{b_n\}_{n=1}^{\infty}$ converges to $\displaystyle B$ and $\displaystyle B \neq 0$, then there is a positive real number $\displaystyle M$ and a positive integer $\displaystyle N$ such that, if $\displaystyle n \geq N$, then $\displaystyle |b_n| \geq M$.
Using the notation from the lemma, what can say about
$\displaystyle M'=\min\{M,~|b_1|,~|b_2|,\cdots,~|b_{N-1}|\}$

3. ## Re: Sequence Question

Originally Posted by Plato
Using the notation from the lemma, what can say about
$\displaystyle M'=\min\{M,~|b_1|,~|b_2|,\cdots,~|b_{N-1}|\}$
That $\displaystyle M'$ is a lower bound of $\displaystyle |b_n|$?

4. ## Re: Sequence Question

Originally Posted by FishStyx
That $\displaystyle M'$ is a lower bound of $\displaystyle |b_n|$?
That is correct.

BUT you need to point out that $\displaystyle M'>0$.
Why is that true?

5. ## Re: Sequence Question

Originally Posted by Plato
That is correct.

BUT you need to point out that $\displaystyle M'>0$.
Why is that true?
I never really gave that much thought...

I just supposed it was stated because $\displaystyle M \leq |b_n| > 0$ since $\displaystyle b_n \neq 0$.

6. ## Re: Sequence Question

Originally Posted by FishStyx
I never really gave that much thought...

I just supposed it was stated because $\displaystyle M \leq |b_n| > 0$ since $\displaystyle b_n \neq 0$.
Well each of these is true, $\displaystyle M>0~\&~(\forall n)[b_n\ne 0]$

7. ## Re: Sequence Question

Originally Posted by Plato
Well each of these is true, $\displaystyle M>0~\&~(\forall n)[b_n\ne 0]$
Hmm, yeah, I can see that.

I'm still not quite seeing the proof yet... Am I supposed to show that there is a positive lower bound on $\displaystyle |b_n|$ for all $\displaystyle n$, specifically $\displaystyle M$ in the context of the problem? Or...