Compare elements of a recursive defined sequence

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The simple reason is that your conjecture is False. Try with:

A = 1
b = 2
c = 1/2

All those numbers are strictly positive, but you will get:

sage: Sequence_rec(3) - Sequence_rec(2)
-20/3

Note also that the denominator can vanish along the loop when i=c+2, which may be another cause of trouble (you will have a lot of poles). By the way, even if the sequence was indeed increasing, Sage will not be able to give an answer for all n together (it does not understands loops symbolically). Moreover, you could even imagine to encode undecidable problems in the iteration of such formulas.

More generally, if you ask for a boolean, you should know that Sage is not able to answer Unknown, hence it will usually answer False when it is not able to prove that answer is True, which is not what mathematicians usually expect.

Actually, there exists an Unknown truth value in Sage but then python will not be able to deal correctly with boolean operations, see

sage: Unknown?

and PEP 335 for more information about this, which was unfortunately rejected, preventing Sage to deal with "proved True", "proved False", "Unable to prove".

If you expect an answer you should tell Sage what are your assumptions (ie in the case you mention assume(c > 2)). Nevertheless, assuming your intution is correct, you can not rely on the output of bool(my_expression) as

sage: assume(c > 3)
sage: bool(Sequence_rec(4) > Sequence_rec(3))
False
sage: bool(Sequence_rec(4) < Sequence_rec(3))
False
sage: bool(Sequence_rec(4) == Sequence_rec(3))
False