# Thread: {a_n}\to A and define {b_n}

1. ## {a_n}\to A and define {b_n}

$\displaystyle \{a_n\}_{n=1}^{\infty}\to A$ and define $\displaystyle b_n=\frac{a_n+a_{n+1}}{2}, \ \ \forall n$

Prove $\displaystyle \{b_n\}_{n=1}^{\infty}\to A$

Is it as simple as $\displaystyle b_n=\frac{A+A}{2}=A$

Therefore, $\displaystyle \{b_n\}_{n=1}^{\infty}\to A$

2. ## Re: {a_n}\to A and define {b_n}

Originally Posted by dwsmith
$\displaystyle \{a_n\}_{n=1}^{\infty}\to A$ and define $\displaystyle b_n=\frac{a_n+a_{n+1}}{2}, \ \ \forall n$

Prove $\displaystyle \{b_n\}_{n=1}^{\infty}\to A$

Is it as simple as $\displaystyle b_n=\frac{A+A}{2}=A$

Therefore, $\displaystyle \{b_n\}_{n=1}^{\infty}\to A$
You need a little more rigor.

Since $\displaystyle a_n \to A$ Pick $\displaystyle N \in \mathbb{N}$ such that for all $\displaystyle n > N$

$\displaystyle |a_n-A|< \frac{\epsilon}{2}$

Now what is

$\displaystyle |b_n-A|=\bigg| \frac{a_n+a_{n+1}}{2}-\frac{2A}{2}\bigg|=...$

3. ## Re: {a_n}\to A and define {b_n}

I don't know why you picked $\displaystyle \frac{\epsilon}{2}$ and I don't know where to go from the $\displaystyle =\cdots$.

Can you elaborate further.

4. ## Re: {a_n}\to A and define {b_n}

Originally Posted by dwsmith
I don't know why you picked $\displaystyle \frac{\epsilon}{2}$ and I don't know where to go from the $\displaystyle =\cdots$.
Actually we could pick just $\displaystyle {\epsilon}$ if you note that
$\displaystyle \left| {\frac{{a_n + a_{n + 1} }}{2} - A} \right| \leqslant \frac{1}{2}\left| {a_n - A} \right| + \frac{1}{2}\left| {a_{n + 1} - A} \right|~.$

5. ## Re: {a_n}\to A and define {b_n}

Originally Posted by dwsmith
I don't know why you picked $\displaystyle \frac{\epsilon}{2}$ and I don't know where to go from the $\displaystyle =\cdots$.

Can you elaborate further.
you can rewrite this difference as the sum of two differences, and use a version of the triagle inequality.....

the "idea" behind this is that any convergent series is Cauchy, so for large enough n, two successive terms are as close as we like (this means something in terms of epsilon), so the "average" of the two successive terms (for large enough n) is going to be close to the limit as well.

6. ## Re: {a_n}\to A and define {b_n}

Originally Posted by Plato
Actually we could pick just $\displaystyle {\epsilon}$ if you note that
$\displaystyle \left| {\frac{{a_n + a_{n + 1} }}{2} - A} \right| \leqslant \frac{1}{2}\left| {a_n - A} \right| + \frac{1}{2}\left| {a_{n + 1} - A} \right|~.$
Ok, so from that, we can say:

$\displaystyle \frac{1}{2}|a_n-A|<\epsilon\Rightarrow |a_n-A|<2\epsilon\Rightarrow |a_n-A|<\epsilon_2$

Therefore, $\displaystyle \{a_n\}\to A$ and the same argument for $\displaystyle a_{n+1}$.

From this, we can then use the definition of b_n with both sequences being A, correct?

7. ## Re: {a_n}\to A and define {b_n}

since {an}-->A, for any ε > 0, there IS some N with |an - A| < ε for all n > N. so 1/2 of that is less than ε/2.

if n > N, surely n+1 > N as well, thus |a(n+1) - A| < ε, too. there is no need for a "secondary" epsilon:

$\displaystyle \left|\frac{a_n + a_{n+1}}{2} - A}\right|\leq\frac{1}{2}|a_n-A|+\frac{1}{2}|a_{n+1}-A| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$