Tricky T/F... convergence T or F?
1) If $x_n \rightarrow 0$ and $x_n \neq 0$ for all $n$, then the sequence {$1/n$} is unbounded.
Also similarly...
2) If {$x_n$} is unbounded and $x_n \neq 0$ for all $n$, then $1/x_n \rightarrow 0$.
For the first one, I would say that is true because the limit of  {$1/n$}  would approach infinity, thus making it unbounded?
And the second one, also seems true by similar logic. Am I overlooking something here?

The first one is true; but note the sequence $(1/x_n)$ need not converge to $\infty$. The sequence $(1/|x_n|)$, however, would.  Consider here, for example, the sequence $(1/2,-1/3,1/4,-1/5,\ldots)$.
For the second one, consider the sequence $(1,1,2,1,3,1,4,1,5,\ldots)$. Note this sequence is unbounded, but the sequence of reciprocals does not converge to $0$. 
(If you knew $(x_n)$ converged to $\infty$ (or to $-\infty$), then you could conclude  the sequence of reciprocals would converge to $0$.)
