Is there any specific approach to prove the divergence of a sequence? For example, I have this problem:
"Prove that the sequence $(a_n)_{n \ge 1}$, where $a_n = n\sqrt2 - [n\sqrt2] + n\sqrt3 -[n\sqrt3]$ is divergent"
(the [ ] stands for the floor function )
I tried to solve it by finding to subsequences that have different limits, but it does not seem to work this way.
Since $\sqrt{2}+ \sqrt{3}$ is irrational, so is $(a_n)$. Consider the first decimal digit of $(a_{10^m})$. If $(a_{10^m})$ would converge, the first decimal digit must eventually become constant, say $d\in\{0,\ldots,9\}$, i.e. for all $m\geq M$ we have first decimal digit of $(a_{10^m})$ equals $d$. But the first decimal digit of $(a_{10^m})$ is the $m$-th decimal digit of $\sqrt{2}+\sqrt{3}$, so that would mean that the irrational number is $d$-periodic, contradiction. So $(a_{10^m})$ doesn't converge and so $(a_{n})$ doesn't converge either.