What happened in Cauchy sequence when the condition " there exist a positive integer" is replaced by " there exist a real number"

Question

My Attempt: In Cauchy sequence and convergent,What happened in Cauchy sequence when the condition " there exist a positive integer" is replaced by " there exist a real number". I think that in Cauchy sequence, the above conditions are interchanged then it is no longer Cauchy. So options b,c discarded. I have no idea about options a and d.

Answer 1

In that question there is no requirement that $n_0$ be an integer. The inequality $n > n_0$ serves only to assert that something happens when the index $n$ is sufficiently large. The status of the four assertions does not depend on that assumption, and would not change if you required it.

Answer 2

As Xander Henderson pointed out in his comment, a counter-example to the proposition is

$$x_n=\sum_{k=1}^{n}\frac{1}{k}.$$

Note that for any $\epsilon>0$ we can set $n_0=\left\lceil\frac{1}{\epsilon}\right\rceil+1$. Then for any $n\geq n_0$

$$|x_{n}-x_{n-1}|=\left|\sum_{k=1}^{n}\frac{1}{k}-\sum_{k=1}^{n-1}\frac{1}{k}\right|=\frac{1}{n}\leq \frac{1}{n_0}<\frac{1}{\left\lceil\frac{1}{\epsilon}\right\rceil}\leq \epsilon.$$

However, it is well known that $x_n$ is the harmonic series and does not converge. Thus, the proposition is false and the answer is d).