For your a) question: note that
Your other question is un readable.
Why not learn to post in symbols? You can use LaTeX tags.
Under the hypothesis that the second sequence is...
(1)
... the difference equation that defines the sequence is...
(2)
The function is illustrated here...
It has a single 'attractive fixed point' in and because is , any 'initial value' will produce a sequence converging at 2 without oscillations...
Kind regards