Help with proofs

Question

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 $q_{n+1}<a$.

Answer 1

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

Answer 2

• $1+\frac{1}{n}$

• $n^2$

• $\pi \cdot n \cdot \left (\cos (\pi(n-1) ))+ 1 \right )$