Does ${x_n:nge1}cup{y_n:nge1}$ has no accumulation point in $X?$












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Let $(x_n),(y_n)$ be two sequences in a metrizable topological space $X$ such that none of $(x_n),(y_n)$ has cluster point in $X.$ Can we then conclude the set ${x_n:nge1}cup{y_n:nge1}$ has no accumulation point in $X?$



I think that should be correct yet I cannot get sure.



Please help.










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    Let $(x_n),(y_n)$ be two sequences in a metrizable topological space $X$ such that none of $(x_n),(y_n)$ has cluster point in $X.$ Can we then conclude the set ${x_n:nge1}cup{y_n:nge1}$ has no accumulation point in $X?$



    I think that should be correct yet I cannot get sure.



    Please help.










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      Let $(x_n),(y_n)$ be two sequences in a metrizable topological space $X$ such that none of $(x_n),(y_n)$ has cluster point in $X.$ Can we then conclude the set ${x_n:nge1}cup{y_n:nge1}$ has no accumulation point in $X?$



      I think that should be correct yet I cannot get sure.



      Please help.










      share|cite|improve this question













      Let $(x_n),(y_n)$ be two sequences in a metrizable topological space $X$ such that none of $(x_n),(y_n)$ has cluster point in $X.$ Can we then conclude the set ${x_n:nge1}cup{y_n:nge1}$ has no accumulation point in $X?$



      I think that should be correct yet I cannot get sure.



      Please help.







      general-topology






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      asked Nov 28 '18 at 2:29









      Jave

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          Yes. Because if $z_0$ is an accumulation point for
          $A={x_n:nge1}cup{y_n:nge1}$ then there is a subsequence of
          $A$ like $(z_n)_{nin mathbb{N}} $ such that $z_nto z_0$. But either the set ${ z_n|nin mathbb{N}} cap {x_n:nge 1} $ have infinte elements or ${ z_n|nin mathbb{N} }cap {y_n:nge1}$ have infinite elements. So there is a subsequence of $(z_n)$ like $(z_{n_k})_{kin mathbb{N}} $ such that $(z_{n_k}) subseteq {x_n:nge1}$(or $subseteq {y_n:nge1} $). But $z_{n_k}to z_0$. So $z_0 $ is an accumulation point for ${x_n:nge1}$(or for ${y_n:nge1}$) and this is a contradiction.






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            Yes. Because if $z_0$ is an accumulation point for
            $A={x_n:nge1}cup{y_n:nge1}$ then there is a subsequence of
            $A$ like $(z_n)_{nin mathbb{N}} $ such that $z_nto z_0$. But either the set ${ z_n|nin mathbb{N}} cap {x_n:nge 1} $ have infinte elements or ${ z_n|nin mathbb{N} }cap {y_n:nge1}$ have infinite elements. So there is a subsequence of $(z_n)$ like $(z_{n_k})_{kin mathbb{N}} $ such that $(z_{n_k}) subseteq {x_n:nge1}$(or $subseteq {y_n:nge1} $). But $z_{n_k}to z_0$. So $z_0 $ is an accumulation point for ${x_n:nge1}$(or for ${y_n:nge1}$) and this is a contradiction.






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              Yes. Because if $z_0$ is an accumulation point for
              $A={x_n:nge1}cup{y_n:nge1}$ then there is a subsequence of
              $A$ like $(z_n)_{nin mathbb{N}} $ such that $z_nto z_0$. But either the set ${ z_n|nin mathbb{N}} cap {x_n:nge 1} $ have infinte elements or ${ z_n|nin mathbb{N} }cap {y_n:nge1}$ have infinite elements. So there is a subsequence of $(z_n)$ like $(z_{n_k})_{kin mathbb{N}} $ such that $(z_{n_k}) subseteq {x_n:nge1}$(or $subseteq {y_n:nge1} $). But $z_{n_k}to z_0$. So $z_0 $ is an accumulation point for ${x_n:nge1}$(or for ${y_n:nge1}$) and this is a contradiction.






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                Yes. Because if $z_0$ is an accumulation point for
                $A={x_n:nge1}cup{y_n:nge1}$ then there is a subsequence of
                $A$ like $(z_n)_{nin mathbb{N}} $ such that $z_nto z_0$. But either the set ${ z_n|nin mathbb{N}} cap {x_n:nge 1} $ have infinte elements or ${ z_n|nin mathbb{N} }cap {y_n:nge1}$ have infinite elements. So there is a subsequence of $(z_n)$ like $(z_{n_k})_{kin mathbb{N}} $ such that $(z_{n_k}) subseteq {x_n:nge1}$(or $subseteq {y_n:nge1} $). But $z_{n_k}to z_0$. So $z_0 $ is an accumulation point for ${x_n:nge1}$(or for ${y_n:nge1}$) and this is a contradiction.






                share|cite|improve this answer












                Yes. Because if $z_0$ is an accumulation point for
                $A={x_n:nge1}cup{y_n:nge1}$ then there is a subsequence of
                $A$ like $(z_n)_{nin mathbb{N}} $ such that $z_nto z_0$. But either the set ${ z_n|nin mathbb{N}} cap {x_n:nge 1} $ have infinte elements or ${ z_n|nin mathbb{N} }cap {y_n:nge1}$ have infinite elements. So there is a subsequence of $(z_n)$ like $(z_{n_k})_{kin mathbb{N}} $ such that $(z_{n_k}) subseteq {x_n:nge1}$(or $subseteq {y_n:nge1} $). But $z_{n_k}to z_0$. So $z_0 $ is an accumulation point for ${x_n:nge1}$(or for ${y_n:nge1}$) and this is a contradiction.







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                answered Nov 28 '18 at 3:36









                Darmad

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