I'm assuming you meant , for the original problem.
It all depends on how rigorous a proof you want. The word "show" has different meanings in different contexts: a pure mathematician means something different by it, sometimes, than an applied mathematician, who means something different from a physicist. If you want to prove, beyond the shadow of a doubt, that the limit is as you've described, you need to do something at least as rigorous as what I've outlined, if not more. If throwing in a few numbers and turning the crank is enough, then just do that. I wouldn't call that a proof by any stretch of the imagination. For one thing, it doesn't take into account the infinite sequence of numbers that are not in the domain of the iteration scheme. Secondly, how do you know it converges to exactly ? A computer has a finite representation of a number with essentially infinite precision. How do you know you're not off in the millionth decimal place?
The lengthy computation I went through does "show" what you're trying to prove. And you might be able to get the subsequence approach to work as well. Either of those is definitely more rigorous than simply plugging in numbers. However, I'm currently working as an engineer, and I know that sometimes the best, most elegant mathematical answer to a problem is not feasible because it takes too long to get. So, if it doesn't work for you, do something else! From my perspective, the long computation or the alternating subsequence approach are the only thing that come to my mind at the moment.