so x(1) := 8, and x(n + 1) = 1/2(X(n)) + 2. show that it is bounded and monotone, and find the limit. so I claim that it is decreasing, i.e, X(n)> x(n+1). by induction: n = 1 implies 8 > 6. this checks. assume K is true, try K + 1. so need to show X(k+1) > x(K+2). we know that X(K+2) = 1/2(x(k+1)) + 2, in othe words, need to show 1/2X(k+1) + 2 < (x(K+1)). bring x(k+1) to other side, then 2 < x(K+1) - 1/2x(k+1). simplify and get 4 <= x(k+1). so now, I know that x(k+1) is less then or equal to 4. this shows that it is bounded by 4 right? I was trying to show that it was decreasing...but I showed this instead. Does this assume that it is decreasing? and how? and the final question. how do I show that the limit is 4?' and question 2: x1 >1, and x(n+1):= 2 - 1/x(n). same thing. now since x 1 is not specifically a number, I dotn know how to approach it. so if I just do soem computations: I know that x1 = something greater then 1, so if I want x2, that is := 2 - 1/(x1), and x3 := 2 - 1/(x2). I'm gonna guess that this will haev a bound of 2. now, since x1 is greater then 1, and it has a bound of 2. upper or lower, my head hurts, so I"m gonna take a break and tackle tihs tommorow...but.....let me think about it.