If the $\lambda \le 1$, then this is easy, otherwise, ftsoc the sequence is bounded then, we have that the averages should converge to some number, which is absurd

kingu wrote: If the $\lambda \le 1$, then this is easy, otherwise, ftsoc the sequence is bounded then, we have that the averages should converge to some number, which is absurd Actually, for a bounded sequence, the averages do not necessarily converge. But we get for $\lambda>1$ the equation $d_{n+1}=\left(\frac{\lambda-1}{n}+1\right)d_n$, where $d_n$ is the average of the first $n$ numbers. So the averages become arbitrarily large, contradiction.

the average is both bounded and increasing in that case, so it will converge