1. Let (xn)n be the sequence given by xn = 1/ n^ 2 + sin(n/6). This sequence...
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Question:
1.
Let (xn)n be the sequence given by xn = 1/ n^ 2 + sin(nπ/6). This sequence has exactly 7 distinct subsequential limits.
a. Find the subsequential limits by finding 7 subsequences (be precise on what the corresponding nk should be), each converging to one of the subsequential limits. (No proof necessary.)
b. Prove (using Theorem 52: A number z is a subsequential limit of a sequence(xn)n
if and only if for everyε >0, the set of indices{n∈N:|xn−z|< ε} is infinite.)
that no other real number can be a subsequential limit
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