Edited-How can I solve polynomial recurrences like $f(n+1)=\frac{2f(n)}{f(n)+1}$

Question

Can anybody tell me the systematic way of solving this recurrence.

$$f(n+1)=\frac{2f(n)}{f(n)+1}$$

I looked over the internet, but could not find the answer. Thanks

{Edit- I am sorry, previously I posted the problem $f(n)=\frac{2f(n)}{f(n)+1}$. Please note that this is not the problem that I intended to ask, and it is not even a recurrence. I have updated my post to reflect the correct problem.}

Now, from what I learnt in high school and college so far, this problem would have been solvable if there was no $f(n)$ in the denominator on the other side. For e.g $f(n+1)=2f(n)$ Then, I would have simply used recursion or Master's Theorem(in case it was applicable). But I do not know how to solve a recurrence like this. I would really appreciate if anybody could point me to the right direction.

Answer 1

We have $$f(n)[f(n)-1]=0$$

$$\implies f(n)=0,1\forall n$$