Not sure on the proof of this:
Prove directly from the definition that the sequence
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.
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 ......