The space of functions $[0,1]^{[0,1]}$ is compact in the topology of pointwise convergence due to Tichonoff's theorem.
Consider the sequence of functions $\{f_n\}_n$ defined by $$f_n(x)=\left\{\begin{array}{} 0 \quad \text{if } x\in \left[\frac{k-1}{2^n},\frac{k}{2^n}\right] \text{ for } k \text { odd in } \{1,\ldots 2^n\} \\ 1\quad \text{otherwise}\end {array}\right.$$
This sequence should have a convergent subnet. Does someone know how to find such a subnet? Thanks!