Not sure on the proof of this:
Prove directly from the definition that the sequence
$\displaystyle (\frac{sin(log(n^{2002})) + 5000}{2n+7})_n$
is cauchy
I will state the following result. Your task is to prove it and show how it applies to your problem. Suppose that $\displaystyle \{a_n\},\{b_n\}$ are sequences such that $\displaystyle 0\leq a_n \leq b_n$. Let $\displaystyle \{ b_n \}$ be Cauchy sequences and $\displaystyle \lim ~ a_n = \lim ~ b_n$. Prove that $\displaystyle \{ a_n \}$ is a Cauchy sequence.
Frankly, I do not know what to make of the original question.
It is well know that a sequence of real numbers converges if and only if it is a Cauchy Sequence.
Take the above hint: Use basic comparison.
If you can show that your sequence converges then it is Cauchy.
Yes ladies and gentlemen (I make no unjustified assumptions here). All of this is what I thought too, but didn't quite feel confident enough with my analysis to suggest it. I was interested to see what others thought before jumping in. A combination of comparison and then the definition would seem to satisfy the boundary conditions.
I quite agree with the puzzle of why his prof won't let the theorem that
a sequence of real numbers converges if and only if it is a Cauchy Sequence
be used. Using this theorem was my first thought.
Unless his prof wants them to practice the whole epsilon thing .....
I suspect the whole metric thing raised by TPH is probably a red herring in this case ......