This problem involves proving statements. You must provide a detailed every term is less than some...

 

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This problem involves proving statements. You must provide a detailed every term is less than some fixed number. A sequence is monotonic if a 2a, for all natural justification for every line of the proof. Do not cut corners. A sequence a,} is bounded above if numbers n. It can be proven in a course in real analysis that if a sequence is monotonic and bounded above then it must converge. Consider the sequence {a,} defined by a, 1 and This problem involves proving statements. You must provide a detalicd a, =l+a if n22. %3D a) Show {a, is bounded by proving every term is less than 2. (hint: assume at least one term is greater than or equal to 2 and get a contradiction.) b) Show {a,} is monotonic. (hint: use mathematical induction. First show that a, 2 4. Then assume that a, 2a, and use that assumption to prove that a24,) c) Now that you have proved that lima, exists, find the limit. Give the exact value, not a decimal approximation. SEVERAL PAGES, JUSTIFY EVERY LINE This problem involves proving statements. You must provide a detailed every term is less than some fixed number. A sequence is monotonic if a 2a, for all natural justification for every line of the proof. Do not cut corners. A sequence a,} is bounded above if numbers n. It can be proven in a course in real analysis that if a sequence is monotonic and bounded above then it must converge. Consider the sequence {a,} defined by a, 1 and This problem involves proving statements. You must provide a detalicd a, =l+a if n22. %3D a) Show {a, is bounded by proving every term is less than 2. (hint: assume at least one term is greater than or equal to 2 and get a contradiction.) b) Show {a,} is monotonic. (hint: use mathematical induction. First show that a, 2 4. Then assume that a, 2a, and use that assumption to prove that a24,) c) Now that you have proved that lima, exists, find the limit. Give the exact value, not a decimal approximation. SEVERAL PAGES, JUSTIFY EVERY LINE