Show that SC (-1,0,1), that is, show that, for all z ER, if ze S (ie....

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Show that SC (-1,0,1), that is, show that, for all z ER, if ze S (ie. z is a cluster point of (r.)), then z€ (-1,0,1) i.e. show the contrapositive: For all x ER, if a {-1,0, 1} then IS. Suppose that z {-1,0,1). (1 pts) Define and explain why e > 0. • Notice that C= min (x-1, , +1} Zak-3-1, 24k-1-1 and 220 as k→ ∞o. (3 pts) Explain why you can say that there is KEN (what should we take for K?), such that, for all k > K Tak-3-1<, ₂2k-0< and 4k-1 +1| < €. (3 pts) Show that there is NE N, such that for n > N, IN-IME (2 pts) Argue that is finite. {n EN: |-|<e} (2 pts) Explain why you can conclude that ii) lim inf (z.)=-1 Notice that (z.) is a bounded sequence. . (2 pts) Explain why and conclude that S=(-1,0,1). lim inf (z.) = inf S lim inf (₂)=-1 iii) lim sup (an) = 1. . (2 pts) Notice that {n} is a bounded sequence and explain why lim sup (n) = sup S and conclude that lim sup (n) = 1. Show that SC (-1,0,1), that is, show that, for all z ER, if ze S (ie. z is a cluster point of (r.)), then z€ (-1,0,1) i.e. show the contrapositive: For all x ER, if a {-1,0, 1} then IS. Suppose that z {-1,0,1). (1 pts) Define and explain why e > 0. • Notice that C= min (x-1, , +1} Zak-3-1, 24k-1-1 and 220 as k→ ∞o. (3 pts) Explain why you can say that there is KEN (what should we take for K?), such that, for all k > K Tak-3-1<, ₂2k-0< and 4k-1 +1| < €. (3 pts) Show that there is NE N, such that for n > N, IN-IME (2 pts) Argue that is finite. {n EN: |-|<e} (2 pts) Explain why you can conclude that ii) lim inf (z.)=-1 Notice that (z.) is a bounded sequence. . (2 pts) Explain why and conclude that S=(-1,0,1). lim inf (z.) = inf S lim inf (₂)=-1 iii) lim sup (an) = 1. . (2 pts) Notice that {n} is a bounded sequence and explain why lim sup (n) = sup S and conclude that lim sup (n) = 1.