Could you guys help me come up with examples of sequences that follow these statements so I can understand how to do these proofs please!!
1) $\forall \epsilon > 0, \exists n > 1$ such that $q_n>1$ and $0<|q_n − q_{n+1}|< \epsilon$.
2) $\forall K > 0, \exists n > 1$ such that $|q_n − q_{n+1}| > K$.
3) For every positive rational number $a$, there exists $n \geq 1$ such that $q_n>a$ and $q_{n+1}<a$.
You need an example of a sequence such that the limit below is true?
1) $q_n = e^{-n} +1 $
2) $q_n = e^n$
3) the statement appears to be incomplete.
And this will help you with formatting:
MathJax basic tutorial and quick reference