Limits, Monotonicity, and Bounding

I have attached the original problem statement.

I'm not entirely sure how to show that the sequence is bounded. The book gave the hint of using the inequality ab<=1/2(a^2+b^2) to show xn+1>sqrt(a) for all n. I'm having trouble applying this inequality however, and am not really sure how to algebraically make use of it. I was looking at defining a and b for the variables, but what I am coming up with isn't making sense to me.

Part b I can do

Part c I was thinking of just working out algebraically by defining xn instead of xn-1 and changing the indexes. Does that sound like the right way to approach it.

I mostly would like some help with part a)

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Originally Posted by renolovexoxo

i have attached the original problem statement.

I'm not entirely sure how to show that the sequence is bounded. The book gave the hint of using the inequality ab<=1/2(a^2+b^2) to show xn+1>sqrt(a) for all n. I'm having trouble applying this inequality however, and am not really sure how to algebraically make use of it. I was looking at defining a and b for the variables, but what i am coming up with isn't making sense to me.

Part b i can do

part c i was thinking of just working out algebraically by defining xn instead of xn-1 and changing the indexes. Does that sound like the right way to approach it.

I mostly would like some help with part a)

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