Uniform convergence and complete metric space

Let $X$ be a metric space and $\{f_n\}$ be a sequence of functions such that $f_n:E\rightarrow X$.

Suppose $f_n\rightarrow f$ uniformly on a set $E$ and $x$ is a limit point of $E$ and $\lim_{t\to x} f_n(t)=A_n$. ($E$ is a subset of some metric space $Y$.)

I know that if $X$ is complete, then $\{A_n\}$ converges. However, i'm really not sure if it is essential. Is it?

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Do you want $E$ to be a subset of some larger space? Or do you want uniform convergence of the $f_n$ on some subset of $E$? – David Mitra Dec 13 '12 at 12:57
@David I meant $E$ to be a subset of arbitrary metric space – Katlus Dec 13 '12 at 13:10

Without the completeness assumption, the statement is false. To see this, take $X=[0,1)$, $Y=[0,1]$, $E=[0,1)$, and for a positive integer $n$, define $f_n:E\rightarrow X$ by $f_n(x) ={n-1\over n} x$. Here you have, for the limit point $1$ of $E$, that $A_n={n-1\over n}$. Now note the sequence $\{A_n\}$ does not converge in $X$.