Course Solutions Uncategorized (Solved) : 15 Given Sequence R R Define S X Set Finite Subsequential Limits X S X Consists Numbers L Q32396753 . . . .

(Solved) : 15 Given Sequence R R Define S X Set Finite Subsequential Limits X S X Consists Numbers L Q32396753 . . . .

 

15. Given a sequence r = (r.), define: ·S(x) as the set of all finite subsequential limits of x, that is, S(x) consists of all numbers l E R such that there exists a subsequence (k n that converges to ·S(x) as the set all subsequential limits of x, that is, S(x) consists of all I E ext (R) RU-oo, +oo) such that there exists a subsequence (k)n that has imit 1 (i) Investigate whether S(x) and S(x) may be empty or not (ii) What are the possible cardinal numbers of S(x) and S(x)? iii) Show that if S(r) is not upper bounded then too E S(x).Similarly, show that if S(x) is not lower bounded then -E S()15.

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