I am having one question that ask finding one sequence defined by $b(n) = \{a_n\} = a_n - [a_n]$ , where $\{x\} = x - [x]$ is a fractional part of a real number $x$ and $[x]$ is the largest integer that is not bigger than $x$, that have one subsequence whose limit is $1$.
We know that $[x] \leq x < [x] + 1$ and apply this we have $b_n$ has to be in the interval $[0,1)$ and then by Bolzano-Weierstrass there exists one subsequence of $b_n$ whose limit is in $[0,1)$ also. But my problem tell me to prove there exist one subsequence of $b_n$ whose limit is in $[0,1]$. So now I have to find a subsequence that converges to $1$. But I have tried to find by expectation with wolframalpha (just for calculating the limit) and still cannot find. If there doesn't exist a subsequence satifying that above condition, I still cannot prove it, just because it relates to arithmetic sequence, I'm not really good at this.
Thank you for your help.