- #1
Mr Davis 97
- 1,462
- 44
I have the following sequence: ##s_1 = 5## and ##\displaystyle s_n = \frac{s_{n-1}^2+5}{2 s_{n-1}}##. To prove that the sequence converges, my textbook proves that the following is true all ##n##: ##\sqrt{5} < s_{n+1} < s_n \le 5##. I know to prove that this recursively defined sequence converges, we have to show that it is decreasing and that it is bounded below. As such, I have a few questions: why does the author choose ##\sqrt{5}## to be the potential lower bound? Would it be just as valid to prove use 0 as the number we could prove to be a lower bound? Also, why is that 5 there? Is it at all necessary to the proof?