# Math Help - Counterexample to uniformly convergence

1. ## Counterexample to uniformly convergence

Hi! My problem is this: Find an example of $(f_n)$, a sequence of functions on $\mathcal{C}(X,\mathbb{R})$ (continuous with domain $X$ and real-valued) such that: $X$ is NOT compact, $(f_n)$ be equicontinuous and pointwise bounded, and every subsequence uniformly convergent have the same limit (call him $f$. In fact, may there's no subsequence uniformly convergent, and we aren't saying that every subsequence is uniformly convergent). I need to find a sequence that satisfy this conditions and NOT converges to f uniformly.

Thanks

Edit: I think that I don't need your help now =) With $f_n(x)=\dfrac{x}{n},\;X=\mathbb{R}$ holds