me a sequeice (unin=1 and Un + an-1 (a) Prove an > 0 for all n EN by induction on n. (b) Prove a? > 2 for all n e N by induction on n. (c) Prove the sequence is decreasing. Tip: Use part (b). (d) Since the sequence is decreasing and bounded below (from part (a)), it necessarily converges (as you showed on the previous problem set). Determine what the sequence converges to. Tip: Use that
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Part c only please.
Note:
The goal amounts to showing that a_n <= a_n/2 + 1/a_n using that it is known (from (b)) that a_n^2 >= 2. multiply both sides by a_n. Note that I am in some sense going the wrong direction here, by assuming what I need to prove. But consider this scratch work --- in the final product, you will want to *start* with a^n_2 >= 2 and divide by a_n to *deduce* a_n <= a_n/2 + 1/a_n.
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