<s>[INST] Argue that $f$ is not uniformly continuous Let $f(x)=x^3$. We want to show that $f$ is not uniformly continuous. 
Can someone please explain from $|f(x_n)-f(y_n)=\big|(n+\frac{1}{n})^3 - n^3)\big| = 3n+\displaystyle\frac{3}{n}$? How does $\big|(n+\frac{1}{n})^3-n^3\big| $ yield $3n+\displaystyle\frac{3}{n}$? ${}{}{}{}$
 [/INST] The first part is just the Binomial Theorem:
$$
    \left(n + \frac{1}{n}\right)^3 = n^3 + 3 n^2 \cdot \frac{1}{n} + 3 n \frac{1}{n^2} + \frac{1}{n^3} = n^3 + 3n + \frac{3}{n} + \frac{1}{n^3}
$$
So
$$
    \left(n + \frac{1}{n}\right)^3 - n^3 = 3n + \frac{3}{n} + \frac{1}{n^3}
$$
The author seems to skip the $\frac{1}{n^3}$ term, but no matter.  The expression is still $>3$, which is the point.
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