2.) If " S " is the set of all "x" such that 0≤x≤1, what points, if any, are points of accumulation of both "S" and C(S)?
3.) Prove that any finite set is closed.
5.) Prove that, if "S" is open, each of its points is a point of accumulation of "S".

1.) Suppose "S" is a set having the number "M" as its least upper bound. If "M" is not a member of "S", show that it is a point of accumulation of "S". Give an example showing that, if "M" does belong to "S" it need not be a point of accumulation of "S".

1.) Suppose that {Xn} is a sequence which is bound and such that all the values X1, X2, X3 .......... are distinct. Assume that the set of these values has just one point of accumulation, denoted by "X". Prove that the sequence is convergent and that the limit is "X".
2.) Consider the sequence with terms 2, 1/2, 4/3, ¼, 6/5, 1/6,.......... Where Xn = ½ [1-(-1) n ] + (1/n). Find two convergent subsequences with different limits.

Solution Summary

Sets and Sequences are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

... We find limits of sequences the same way that we find ... are getting smaller and 1 is the largest number in the set. ... in (a), we need to consider the sequence {1/n ...

... Fundamental mathematics sequences are examined in the solution. ... (1). (a). The sequence elements are: ... Without loss of generality set n m then if we require: ...

... coding sequence. ... the density of HCNEs in the intronic and intergenic sequences ﬂanking every ... 10-fold enrichment for HCNEs relative to the full set of 1,285 ...

... We set A = F (f ) and suppose {tn }∞ is a sequence of points ... continuous if and only if tn → t ⇒ f (tn ) → f (t) for all sequences {tn }∞ with ...

... Remarks so far: About : Theorem 2' on page 48 says that a point is a limit point of a set if and only if there is a sequence of set elements converging to it. ...

... a protein that is translated in the ER is modified through cleavage of the signal sequence and often cleavage of pro-domains that require an in vivo set up to ...

... Would this code sequence compile? Explain. (5 points) Answer: 17. What does the following pseudocode do? (4 points). Input x Input y Set result = (x+y)/2 Output ...

... theory that might help explain how all stakeholders within that setting make decisions ... theory also presents a dynamic process, not a fixed sequence, it seems to ...

... gives detailed steps on setting up likelihood function, finding MLE and then discussing properties. It involves finding the likelihood of any sequence of s ...

Statistical Inference Problem Set. Please help with the following problems. (i) Let {Xi} be a sequence of rv's that converges in probability to a constant a ...