I understand that no matter what value $a_0$ takes on between $0$ and $1$ that $a_1\leq \frac{1}{2}$. This has lead me to believe that for all $n\geq 1$, $b_n=(\frac{1}{n}-\frac{1}{n^2})<a_n <\frac{1}{n}=c_n$. If this can be proven true, then it is easy to see, by the Squeeze/Sandwich Theorem that $\lim_{n\rightarrow\infty}nb_n\leq\lim_{n\rightarrow\infty}na_n\leq\lim_{n\rightarrow\infty}nc_n$ and since $(nb_n)=(1-\frac{1}{n})\rightarrow1$ and $(nc_n)=(1)\rightarrow1$, then $(na_n)\rightarrow1$.
If I am completely off-base in these ideas, let me know too. Thanks.
