Let $x_n$ be a an unbounded sequence of non-zero real numbers. Then
it must have a convergent subsequence.
it can not have a convergent subsequence.
$\frac{1}{x_n}$ must have a convergent subsequence.
$\frac{1}{x_n}$ can not have a convergent subsequence.
Well, 1. is not true as say $x_n=n\ \forall n$, 2. I am not sure, but I guess there may be an unbounded sequence which may have an convergent subsequence, 3. is true due to Bolzano-Weierstrass as it is bounded sequence, 4. is false.
Am I right? Please correct me if I am wrong anywhere. Thank you.