Exercise 2.2.7. Here are two useful definitions: (i) A sequence (an) is eventual
ID: 3144678 • Letter: E
Question
Exercise 2.2.7. Here are two useful definitions: (i) A sequence (an) is eventually in a set A C R if there exists an N e N N. (ii) A sequence (an) is frequently in a set A C R if, for every N such that an A for all n N, there exists an n 2 N such that an E A (a) Is the sequence (-1)" eventually or frequently in the set (1)? (b) Which definition is stronger? Does frequently imply eventually or does eventually imply frequently? quently or eventually. Which is the term we want? to 2. Is (n) necessaril (c) Give an alternate rephrasing of Definition 2.2.3B using either fre- (d) Suppose an infinite number of terms of a sequence (an) are equal .1)? Is it y eventually in the interval (1.9, 2 frequently in (1.9,2.1)? ercise 2.2.8. For somAA1t.Explanation / Answer
Hi,
Given sequence is (-1)^n i,e it contains only {-1,1} when n is odd its -1 and 1 when n is even.
since the given set is R, there are many many n for which sequence is -1 and same with case of 1.
Therefore there is no such N such that (-1)^n is 1 for evey n>=N
But for every N, there exists a n>=N such that (-1)^n = 1, simpley choose like the following
if N is odd choose N+1
if N is even choose N+2
hence sequence is frequently in set {1} and NOT eventually
b. Eventually implies frequently, because by definition of eventually, for all n>=N an is in some set S, i,e there is K for every n>=K an is eventually, we just have to choose max(N,K) then it implies there exists an n>=max(N,K), however vice versa is not true
c. 2,2.3B question not given
d. Here this is almost same as the first sub part, here xn={2},
now, since the interval is {1.9,2.1} hence there could be a number for which the 1.9 is not in {2} therefore for all is wrong, but there exists an n>=N such that xn={2}
hence frequently .
Thumbs up if this was helpful, otherwise let me know in comments.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.