Yes I believe thats ok... Its induction isnt it?

Well there could be different ways to the limit, i tried the hardest way out there. I actually found out the sequence explicitly

There are different approaches, one of them is using generating functions and the other is by redefining the sequence and simple algebraic tricks.

I will do it using generating functions,

Let

This means

But for our series, . And .

Thus,

Clearly both can be represented as geometric progressions. And also the above function exists for some values of x(its called the region of convergence)

Thus:

By polynomial equality,

Now very clearly, .