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

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October 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 {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].
October 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 {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 $(a_n)\to L$ then $\left( {\frac{1}{n}\sum\limits_{k=1}^n {a_k } } \right) \to L$
Using that we can define $a_n=b_n-b_{n-1}-b$ (if necessary define $b_0=0$ ).
Now clearly $(a_n)\to 0$ so $\left( {\frac{1}{n}\sum\limits_{k=1}^n {a_k } } \right) \to 0$ .
But $\left( {\frac{1}{n}\sum\limits_{k = 1}^n {a_k } } \right) = \frac{{b_n }}{n} - b$ .
Because you missed class and because I have no idea about Th3, that proof may not work.
October 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

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.
October 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
b_n/n → b as n → ∞

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

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