I am currently reading Baby Rudin and I am having trouble understanding why convergent sequences must have a bounded range. Specifically, I am thinking of the following counterexample: $f(n)=1/(n-1)$
At $n=1$, this sequence diverges, but at infinity, it "converges" to zero. Any insight into the above theorem and my "counterexample" would be appreciated. Also, I apologize for the lack of $\LaTeX$. I am doing this from my phone.