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Math Help - True or False Questions for Real Analysis

  1. #1
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    True or False Questions for Real Analysis

    True or False, if false give a counter example.

    If S is a set of complex numbers and C is a positive real number such that |s| < C for all s in S then there exists a smallest circle in the complex plane which contains all the elements in S.

    Every convergent series in a Metric Space is Cauchy.

    Every Cauchy series in a Metric space is convergent.

    If a series passes the root test, it converges absolutely

    If (a_n) is a sequence of positive reals which tends to 0, then \sum_{0, infinity} ((-1)^n)a_n converges

    A subspace of a metric space is compact if it is closed and bounded

    Suppose f:X->Y is a continuous map of a metric space, and X is compact. Then if U is a closed subset of X, f(U) is a closed subset of Y.

    If f and g are two real valued functions on R and both g and fg are continuous, then f is continuous.

    Thanks in advance for your help, I'll try to change the list as I answer them.
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  2. #2
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    Quote Originally Posted by terr13 View Post
    True or False, if false give a counter example.

    If S is a set of complex numbers and C is a positive real number such that |s| < C for all s in S then there exists a smallest circle in the complex plane which contains all the elements in S.

    Every convergent series in a Metric Space is Cauchy.

    Every Cauchy series in a Metric space is convergent.

    If a series passes the root test, it converges absolutely

    If (a_n) is a sequence of positive reals which tends to 0, then \sum_{0, infinity} ((-1)^n)a_n converges

    A subspace of a metric space is compact if it is closed and bounded

    Suppose f:X->Y is a continuous map of a metric space, and X is compact. Then if U is a closed subset of X, f(U) is a closed subset of Y.

    If f and g are two real valued functions on R and both g and fg are continuous, then f is continuous.

    Thanks in advance for your help, I'll try to change the list as I answer them.

    1.no, the set must be a compact set(bounded and closed), otherwise there is no smallest circle.

    2.yes

    3.no, the metric space must be complete.

    4.yes

    5.yes

    6.yes

    7.yes

    8.your statement is not clear, is fg=f(x)g(x), or fg=f(g(x))?
    Last edited by frankmelody; October 21st 2008 at 01:53 PM. Reason: a little chage to 7
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