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].
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 , we have .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 or."
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.