Suppose we are given with a sequence of nonnegative terms denoted by $\{a_n\}$ and a recurrence relation satisfied by the sequence $$a_{k+1}-a_k \leq F_k a_k^{\frac{2}{1+\epsilon}} $$ where $ \epsilon > 0$, $ F > 0$.
Suppose I don't know whether the limit of above sequence exists or not, and I want to find out the limit. So for that, I suppose that limit of this sequence is $a$ and from the recurrence relation, I get the value of $a=0$. So, Is it necessary to show that limit exists before calculating it or the value of limit itself implies its existence?