I just received my first assignment for a mathematical proofs course I am taking this year. We just began the course, and we have so far only covered examples of proofs (how to prove if-then statements in different ways) and the mathematical principle of induction.
Here is the question I am having difficulty with:
"Define a sequence $a_n, n \ge 0,$ inductively by $a_0 = 2,$ and for all $n \ge 0, a_{n+1} = \sqrt{a_n + 1}.$
a) Prove that for every $n \ge 0, a_n > \frac{1+\sqrt{5}}{2}.$
b) Prove that for every $n \ge 0, a_n > a_{n+1}.$ (You may use the fact that the polynomial $x^2 - x - 1 < 0$ if and only if $\frac{1-\sqrt{5}}{2} < x < \frac{1+\sqrt{5}}{2}.$
What would be the best proof technique for these questions? Should I prove both using induction, or is there a simpler/better way?