Does $\limsup_{n \to \infty} x_n = 1$ imply $(x_n^n)_{n=1}^\infty$ is bounded?
Solution 1:
There are sequences with limit superior equal to 1 where $x_n^n$ is unbounded. Example: let $x_n=1+\frac12$ for the first few $n$, until $x_n^n>10$. Let the next few $x_n=1+\frac13$ until $x_n^n>100$. Let the next few $x_n=1+\frac14$ until $x_n^n>1000$. Repeat the pattern. Now $x_n$ converges to 1, so has $\limsup x_n=1$, but $x_n^n$ is unbounded.