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Let X be a topological space and let Y be a metric space

  1. Dec 8, 2008 #1
    Hello.

    Please, help me with this exercise:

    Let X be a topological space and let Y be a metric space. Let [tex]f_n: X \rightarrow Y[/tex] be a sequence of continuos functions. Let [tex]x_n[/tex] be a sequence of points of X converging to x. Show that if the sequence [tex](f_n)[/tex] converges uniformly to [tex]f[/tex] then [tex](f_n(x_n))[/tex] converges to f(x).

    Thanks in advance.
     
  2. jcsd
  3. Dec 8, 2008 #2
    Re: Functions

    What attempts have you made at this problem?
     
  4. Dec 9, 2008 #3
    Re: Functions

    Okay, here it is.
    I don't know if this is correct...

    Let [tex]A \in \tau[/tex] such that [tex]f(x)[/tex] [tex]\in A[/tex], then, [tex]f^-^1(A)[/tex] [tex]\in \tau[/tex] and [tex]x[/tex] [tex]\in[/tex] [tex]f^-^1(A)[/tex]

    Since [tex]x_n[/tex] [tex]\rightarrow[/tex] [tex]x[/tex] implies that [tex]\exists k \in \mathbb N[/tex] such that [tex]x_n[/tex] [tex]\in[/tex] [tex]f^-^1(A)[/tex] [tex]\forall[/tex] [tex]n \succ k[/tex]



    or


    given [tex]\epsilon \succ 0[/tex], [tex]\exists k \in \mathbb N[/tex] such that [tex]d(x_n, x)[/tex] [tex]\prec \epsilon[/tex]


    Besides, [tex](f_n)[/tex] converges uniformly to [tex]f[/tex] i.e. [tex](f_n)\stackrel{u}{\rightarrow} f[/tex], implies that given [tex]\epsilon \succ 0[/tex], [tex]\exists k \in \mathbb N[/tex] such that [tex]d(f_n(x), f(x))[/tex] [tex]\prec \epsilon[/tex] [tex]\forall[/tex] [tex]n \succ k[/tex]



    I think, I have to show that exists [tex]m \in \mathbb N[/tex] such that [tex]f_n(x_n)[/tex] [tex]\in A[/tex] [tex]\forall [/tex] [tex]n \succ m[\tex]

    I don't know how to do that.
    I need help.
    This is all I have. Thanks in advance.
     
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