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I am reviewing for a final exam and can't seem to figure out how to do this one. This was a question from an exam this semester.

Suppose that $\lim_{n \rightarrow \infty} x_n = 0$.

Prove that there exists $N \in \mathbb{N}$ such that $\left|x_n^2 + x_{n+3}\right| < \frac{1}{5}$ for all $n \geq N$.

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Since $x_n \to 0$, then there exists an integer $N$ such that $$ |x_n|<\frac1{10}, \quad n\geq N, $$ and we have $$ |x_n^2+x_{n+3}|\leq|x_n^2|+|x_{n+3}|<\frac1{10}\times\frac1{10}+\frac1{10}<\frac15, \quad n\geq N. $$

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Since $\displaystyle\lim_{n\to\infty}x_n=0$, $\;\;\displaystyle\lim_{n\to\infty}(x_n^{2}+x_{n+3})=0$,

so there is an $N\in\mathbb{N}$ such that $|x_n^{2}+x_{n+3}|<\frac{1}{5}$ for $n\ge N$.

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