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Math Help - Topolgy sequence

  1. #1
    MHF Contributor Amer's Avatar
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    Topolgy sequence

    I'm reading about sequences in Topology and I do not know how to solve this

    let R^1 have the left ray topology

    a) dose the sequence f:N\longrightarrow R^1 given by f(n)=\frac{1}{n} converges if so to what point or points

    from definition f converges to a\in R^1 if far all U open in left ray topology and a\in U there exist n_o\in N such that if n>n_o then f(n)\in U

    ok if we take 1<= x in R and take U contains x take n_o=1 so any n>n_o we have f(n)\in U

    so f converges to all x>=1 is this right or not
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  2. #2
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    Quote Originally Posted by Amer View Post
    I'm reading about sequences in Topology and I do not know how to solve this
    let R^1 have the left ray topology
    I think you need to explain left ray topology.
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  3. #3
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Plato View Post
    I think you need to explain left ray topology.
    I know left ray topology, the open sets in left ray topology (-\infty,a) such that a\in R
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  4. #4
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    I think it converges to every element of \mathbb{R} ^+. Clearly every open set that contains a>0 contains every element b<a so it has the tail of f(n) so f converges to a. If we take a\leq0 then clearly (-\infty,a) does not contain any element f(n) so f converges only to those in \mathbb{R} ^+
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