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Math Help - a sequence of real numbers

  1. #1
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    a sequence of real numbers

    Let
    {xn} be a sequence of real numbers that is bounded above by M and such that

    xn x . Prove that x M .


    we know that
    A
    sequence {xn} is bounded if there exists an x X and a real number

    M > 0 such that d (xn , x) M for all n . also it convergent to x.

    how can I use this to prove. any help will be apprciated.
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  2. #2
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    Quote Originally Posted by koko2009 View Post
    Let











    {xn} be a sequence of real numbers that is bounded above by M and such that

    xn x . Prove that x M .


    we know that

    A
    sequence {xn} is bounded if there exists an x X and a real number
    M > 0 such that d (xn , x) M for all n . also it convergent to x.

    how can I use this to prove. any help will be apprciated.
     x>M \Longrightarrow\,\, choose\,\, e=\frac{x-M}{2}\Longrightarrow |x_n-x|<e\,\,\, \forall n>N\,,\,\,for\,\,some\,\,N \in \mathbb{N} \Longrightarrow \frac{M-x}{2}= -e \leq x_n-x \leq e=\frac{x-M}{2}

    Now get your contradiction that \forall n \in \mathbb{N}\,,\,|x_n|<M

    Tonio
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