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 {b_{n}} be a sequence in IR with the property that b_{n+1} − b_{n} → b as

n → ∞. Prove that b_{n}/n → b as n → ∞. [Hint: Use Theorem 3 from

class].

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 {b_{n}} be a sequence in IR with the property that b_{n+1} − b_{n} → b as

n → ∞. Prove that b_{n}/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.

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

b_{n}/n → b as n → ∞

where the b_{n} is a term of the ORIGINAL {b_{n}} 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 (*x*_{n}) 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 (*x*_{n}) 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.

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

b_{n}/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 $\displaystyle \left( {\frac{{b_n }}{n}} \right) \to b$