Math Help - a_{n+1}=a_n+1/S_n - sequence, limit

1. a_{n+1}=a_n+1/S_n - sequence, limit

$a_1=1$, $a_{n+1}=a_n+\frac 1{S_n}$ where $n=1,\,2,\ldots$ and $S_n=a_1+a_2+\ldots +a_n$. Prove that $\lim_{n\rightarrow\infty}a_n=\infty$. Take $b_n=n(a_n^2-a_{n-1}^2)$; does it have a limit if yes what?

2. Originally Posted by james_bond
$a_1=1$, $a_{n+1}=a_n+\frac 1{S_n}$ where $n=1,\,2,\ldots$ and $S_n=a_1+a_2+\ldots +a_n$. Prove that $\lim_{n\rightarrow\infty}a_n=0$.
That is impossible. $\left(a_n\right)$ is clearly increasing and each term is $\ge 1$.

3. Originally Posted by Plato
That is impossible. $\left(a_n\right)$ is clearly increasing and each term is $\ge 1$.
I'm sorry. My bad. Corrected it!