Limit of $a_{n+1}= \frac{n}{n+1} a_n$

Question

I think that this sequence $$a_{n+1}= \frac{n}{n+1} a_n$$

can be rewritten as $$a_n= \frac{1}{n+1}a_0.$$ Therefore the limit should be $0$.

But my proof by induction turns out wrong. Is my idea correct? And how can I prove it?

Answer 1

Let $na_n=b_n$. Then, one has $$a_{n+1}=\frac{n}{n+1}a_n\Rightarrow (n+1)a_{n+1}=na_n\Rightarrow b_{n+1}=b_n.$$ So, $b_n=b_1=a_1$. Hence, $na_n=a_1$, i.e. $a_n=\frac{a_1}{n}$.