Let $(x_n)_n$ be an increasing sequence of positive integers. Can we find a positive integer $j$ so that the new sequence $(y_n)_n=(x_n - j)_n$ has a sub-sequence consisting of perfect squares?
I ended up this question while thinking about the strong law of large numbers. I was able to show almost sure convergence for sub-sequences with $(k^2 +j)_k $ type indices, and I am trying to extend the almost convergence result for the original sequence.
Any kind of hint or answer is appreciated!
Not necessarily. We can take $(x_n)_n$ to be a sequence that gets farther and farther from perfect squares. For example, take $x_n = n^2+n$. Take any positive integer $j$. Then, if $n > j$, $x_n - j = n^2+(n-j)$ is not a perfect square.