Since, $\mathbb Q\cap[0,1]$ is an countably infinite set, then there exists a bijection (enumeration) from $\mathbb N$ to $\mathbb Q\cap[0,1]$, which gives us a sequence in $\mathbb Q\cap[0,1]$.
If $(x_n)_n$ is a sequence of enumeration of $\mathbb Q\cap[0,1]$, then it can be easily verified that this sequence is not convergent in usual sense.
But my question is about the almost convergence and statistically convergence of the above sequence.
My Question : If $(x_n)_n$ is a sequence of enumeration of $\mathbb Q\cap[0,1]$, then is it statistically convergent? Also, what is about almost convergent here?
Consider the following enumeration of $\mathbb Q\cap [0,1]$: $$ \Bigl(\bigl(\frac{i}{n}\bigr)_{i=0}^n\Bigr)_{n=1}^{\infty} $$ (Note that the fractions are not reduced, so it is not a bijective enumeration - but this is a relatively small detail.)
For every $x\in[0,1]$ and $\epsilon\in (0,1/2)$, the set of $(i,n)$ with $|x-i/n|<\epsilon$ has density $2\epsilon$, since for all $n$ sufficiently large, there are roughly $2\epsilon n$ values of $i$ satisfying the inequality for the given $n$. (Something similar should also occur for the variant of this enumeration with duplicate fractions removed.) Thus, this particular enumeration does not converge statistically to any number.
I believe the same sequence also provides a counterexample to the almost convergence question, as well.