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.

... Convergent Sequences and Subsequences and Accumulation Points. Prove that a set A, a subset of the real numbers, is compact if and only if every sequence {an ...

... 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 ...

... take any Si, any ei, {xi} is a sequence with limit ... X which is different from p. Prove without using sequences. ... of open set, open interval, and that the point p ...

... 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: ...

... 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 ...