Sequences (Induction?) Problem

[bz89]

Homework Statement
Consider the sequence {a_n} where a₁ = sqrt(k), a_n+1 = sqrt(k + a_n), and k > 0.

a. Show that {a_n} is increasing and bounded.

b. Prove that the limit as n approaches infinity of a_n exists.

c. Find the limit as n approaches infinity of a_n.

The attempt at a solution

b is straightforward. If you show that a_n is monotonic and bounded then it has a limit.

I don't really understood how to approach a. The solutions guide suggests some sort of induction that starts with a_n <= ((1 + sqrt(1 + 4k))/2). I don't understand how I would be able to go from the givens to that point.

 

[dirk_mec1]

a) calculate a(n+1)-a(n) and draw your conclusion.

b)-

c) In the limit for n tending to infinity you'll get: $$L = \sqrt{k+L}$$ which you can solve.