convergence divergence - Prove that the sequence: $a_1 = 1, a_{n+1} =sqrt{c+da_n}$ (when the real numbers $c, d > 1$) is converging and find it's limit

I have a summarized solution but it's starts with proving that the sequence is bounded from above by c+d.
How can I know that this sequence is bounded by c+d? I understand the proof by induction but how do I actually realize that fact?

I know how to prove that the sequence is monotonically increasing and to find it's limit after we've established that the sequence is converging.