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.

... s_1, s_2, ..., s_n, ...) be a countably infinite sequence of 0's ... to-one correspondence between P(A) and the set S of all countably infinite sequences of 0's ...

... 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 ...

... c sequence and must be bounded above or bounded below. Clearly, xn < xn+1 < x in (a, b) => f (xn) < f (xn+1), f is increasing and bounded in the open set (a, b ...

... 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 ...

... are almost everywhere the limit of a non-decreasing sequence of step functions for which the sequence (∫cn) is bounded. 3. Show that ¼ is in the Cantor set. ...

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

... for encoders: Encoding a given sequence of octets to different, equivalent encoded sequences is perfectly ... for text in certain character sets, while such ...

... Looking at a 3-bit counting sequence: ... Note that in order to set Q to low regardless of the CL or D inputs we need to supply both Low CLR (the input is inverted ...