Convergence of the sequence $\left(\frac{1}{n}\right)^{\sin\left(\frac{1}{n}\right)}$ I tried using the logarithmic method. But I couldn't work with it as I am not sure we can apply L'Hospital to the indeterminate form -infinity/infinity.
Please help me prove the convergence.
For any $x\in(0,1)$ we have $x\sin(1) \leq \sin(x) \leq x$ by concavity, hence
$$ \lim_{n\to +\infty}\frac{1}{n^{\sin(1/n)}} = \lim_{n\to +\infty}\sqrt[n]{1/n}=1 $$
holds by squeezing.
use that $$|\sin(1/n)|\le 1$$ is hold