Define a sequence s(sub n) by s(sub 1) = 1, s(sub n+1) = (s(sub n) )+ (s(sub n)/(1+s(sub n)^2)) for n greater than/equal to one. Show this sequence is unbounded.
Suppose that it does converge, to a limit $\displaystyle l$. Let $\displaystyle n\to\infty$ in the recurrence relation $\displaystyle s_{n+1} = s_n + s_n/(1+s_n^2)$, and you see that $\displaystyle l = l + l/(1+l^2)$. Therefore $\displaystyle l=0$. But that is impossible, because $\displaystyle s_1=1$, and the sequence is increasing, so its limit cannot be less than 1.