# Limit of sequence {b_n}/n --> b

• Oct 23rd 2012, 01:29 PM
mickyduu
Limit of sequence {b_n}/n --> b
This should be easy, but I missed a week of my real analysis class and the problem recommends using a theorem not in the text.

Let {bn} be a sequence in IR with the property that bn+1 − bn → b as
n → ∞. Prove that bn/n → b as n → ∞. [Hint: Use Theorem 3 from
class].
• Oct 23rd 2012, 02:36 PM
Plato
Re: Limit of sequence {b_n}/n --> b
Quote:

Originally Posted by mickyduu
This should be easy, but I missed a week of my real analysis class and the problem recommends using a theorem not in the text.
Let {bn} be a sequence in IR with the property that bn+1 − bn → b as
n → ∞. Prove that bn/n → b as n → ∞. [Hint: Use Theorem 3 from
class].

Shame on you for missing class. It would be nice to know what is th. 3.
There is a theorem of means:
If $\displaystyle (a_n)\to L$ then $\displaystyle \left( {\frac{1}{n}\sum\limits_{k=1}^n {a_k } } \right) \to L$
Using that we can define $\displaystyle a_n=b_n-b_{n-1}-b$ (if necessary define $\displaystyle b_0=0$).
Now clearly $\displaystyle (a_n)\to 0$ so $\displaystyle \left( {\frac{1}{n}\sum\limits_{k=1}^n {a_k } } \right) \to 0$.
But $\displaystyle \left( {\frac{1}{n}\sum\limits_{k = 1}^n {a_k } } \right) = \frac{{b_n }}{n} - b$.
Because you missed class and because I have no idea about Th3, that proof may not work.
• Oct 23rd 2012, 03:08 PM
mickyduu
Re: Limit of sequence {b_n}/n --> b
Thanks for the help, but I don't think that your answer works, or more precisely I don't think that it answers the question. I'm pretty sure that I need to show that the

bn/n → b as n → ∞

where the bn is a term of the ORIGINAL {bn} sequence.

My guess is that I'm going to need to start with the definition of a limit for a convergent sequence:

"A real number x is the limit of the sequence (xn) if the following condition holds:for each ε > 0, there exists a natural number N such that, for every http://upload.wikimedia.org/math/1/7...9574566b34.png, we have http://upload.wikimedia.org/math/0/1...3b40c5e581.png.In other words, for every measure of closeness ε, the sequence's terms are eventually that close to the limit. The sequence (xn) is said to converge to or tend to the limit x, written http://upload.wikimedia.org/math/4/6...96553475ef.png orhttp://upload.wikimedia.org/math/1/5...ca9eb2d60a.png."

I believe that theorem 3 is going to be a common analysis theorem, possibly one that is better known in some generalized form or when dealing with functions2nd +, however our text merely glosses over dealing with sequences.

• Oct 23rd 2012, 03:15 PM
Plato
Re: Limit of sequence {b_n}/n --> b
Quote:

Originally Posted by mickyduu
Thanks for the help, but I don't think that your answer works, or more precisely I don't think that it answers the question. I'm pretty sure that I need to show that the
bn/n → b as n → ∞

Yes it does. PLease see my EDIT. There was a cut&paste error.

It proves that $\displaystyle \left( {\frac{{b_n }}{n} - b} \right) \to 0$ so $\displaystyle \left( {\frac{{b_n }}{n}} \right) \to b$