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Math Help - Limit Superior

  1. #1
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    Limit Superior

    Suppose that \displaystyle{x_n} is a bounded sequence of real numbers.Show that \displaystyle{y_n=\sup_{m \geq n}x_m} is a bounded decreasing sequence and deduce that \displaystyle{y_n \longrightarrow y},ie \displaystyle{\limsup_{n\to\infty}x_n=y}.
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  2. #2
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    Quote Originally Posted by Markeur View Post
    Suppose that \displaystyle{x_n} is a bounded sequence of real numbers.Show that \displaystyle{y_n=\sup_{m \geq n}x_m} is a bounded decreasing sequence and deduce that \displaystyle{y_n \longrightarrow y},ie \displaystyle{\limsup_{n\to\infty}x_n=y}.
    If N_1< N_2, the \{x_m| m\ge N_2\} is a subset of \{x_m| m\ge N_2\}. Any upper bound for the second is an upper bound for the first.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    If N_1< N_2, the \{x_m| m\ge N_2\} is a subset of \{x_m| m\ge N_2\}. Any upper bound for the second is an upper bound for the first.
    Hmm you mean \{x_m| m\ge N_2\} is a subset of \{x_m| m\ge N_1\}?And how to prove that the sequence is decreasing as well?
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