If $\,S_n/n\,$ converges to a finite limit $c$, why does $\,\left(S_{n+1}-S_{n}\right)/n\,$ converge to zero?

Question

If we have a sequence $S_n$ and know as a fact that $S_n/n$ converges to some finite limit $c$, why is it true that $(S_{n+1}-S_n)/n$ converges to zero? I could see this for $n+1$ in the denominator but not for $n$. Because I am not convinced that $S_{n+1}/n$ converges to $c$ necessarily.

I would greatly appreciate your help.

Answer 1

Hint:

Since $$\frac{S(n+1)-S(n)}{n}=\frac{S(n+1)}{n}-\frac{S(n)}{n}=\frac{n+1}{n}\frac{S(n+1)}{n+1}-\frac{S(n)}{n}$$ we have \begin{align*} \lim_{n\to \infty}\frac{S(n+1)-S(n)}{n}&=\left[\lim_{n\to \infty}\frac{n+1}{n}\right]\left[\lim_{n\to \infty}\frac{S(n+1)}{n+1}\right]-\lim_{n\to \infty}\frac{S(n)}{n} \end{align*}

Answer 2

Observe that $$ \frac{S_{n+1}}{n+1} = \frac{S_{n+1}}{n} \frac{n}{n+1}, $$ and also that $$ \frac{n}{n+1}= \frac{n+1-1}{n+1}=1-\frac{1}{n+1} \underset{n\to +\infty}{\to} 1. $$