The question I have to prove is the following:
Let $D(x)$ be Dirichlet Function:
$$D(x) = \begin{cases}1 & x\in \Bbb Q \\ 0 & x \notin \Bbb Q \end{cases}$$
Let $(a_n)_{n=1}^{n\to\infty}$ be a convergent sequence.
Determine if $(a_n) + D(a_n)$ is convergent.
I have a feeling that I can find an example the contradicts it, but I wasn't able to. Maybe The statement is correct?
Thanks,
Alan
Let $L=\lim a_n$.
If the sequence $a_n$ has finitely many rational numbers, $$\lim [a_n+D(a_n)]=L+1$$
If it has finitely many irrational numbers, $$\lim [a_n+D(a_n)]=L$$
Otherwise, $a_n+D(a_n)$ does not converge, because it has a subsequence that converges to $L$ and another subsequence that converges to $L+1$.