I came across the following problem during the course of my study of real analysis:
Prove that $(x_n)$ is a null sequence iff $(x_{n}^{2})$ is null.
For all $\epsilon>0$, $|x_{n}| \leq \epsilon$ for $n > N_1$. Let $N_2 = \text{ceiling}(\sqrt{N_1})$. Then $(x_{n}^{2}) \leq \epsilon$ for $n > N_2$. If $(x_{n}^{2})$ is null then $|x_{n}^{2}| \leq \epsilon$ for $n>N$. Let $N_3 = N^2$. Then $|x_n| \leq \epsilon$ for $n> N_3$.
Is this correct? In general, we could say $(x_{n})$ is null iff $(x_{n}^{n})$ is null?