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 $$f_n(x_n)$$ $$\in A$$ $$\forall$$ $$n \succ m[\tex]<br /> <br /> I don't know how to do that.<br /> I need help.<br /> This is all I have. Thanks in advance.$$