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

[seed21]

Hello.

Please, help me with this exercise:

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

Thanks in advance.

 

[adriank]

What attempts have you made at this problem?

 

[seed21]

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

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

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



or


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


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



I think, I have to show that exists m \in \mathbb N such that f_n(x_n) \in A \forall n \succ m[\tex]<br /> <br /> I don&#039;t know how to do that.<br /> I need help.<br /> This is all I have. Thanks in advance.