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 ...

... 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 ...

... Then, we can construct a sequence of rational numbers defined by a finite decimal fraction. 0. ... Recall that E is a set of infinite sequences made of 0 and ...

... The initial state can be anything but it is convenient to set it to start at '00000'. For example if the input sequence is 010101f, the machine will work ...

Genetics Problem Set. ... at Arg since it's the first and last amino acid in the sequence. ... The strand that has any of these base sequences at both the beginning ...

... 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 ...