Suppose a sequence of positive terms is given satisfying some recurrence relation.
I want to check whether the sequence is convergent or not.
For this, I assumed in the beginning that all the iterates of the sequence lie in some neighborhood of a given point and in this case I am able to prove the convergence of the sequence.
Now, can I say that my assumption, in the beginning, holds? Means on proving the convergence, can I say back that the iterates lie in some neighborhood of a given point
The status of the convergence could depend on the initial terms. For some values of the initial terms, you may be able to prove that the sequence converges, whereas it would diverge for other values. Therefore, proving the convergence with an additional assumption does not allow you to go back and say that this assumption has to hold in every case.
Think of $$u_n=a^n$$ where $a$ is a positive number. This will tend to $0$ if $a$ is small and to infinity if $a$ is larger than $1$.
You could argue that this is not a recurrence relation, however you can easily make it so:
$$u_{n+1}={u_n}^{\frac{n+1}{n}}$$