<s>[INST] Prove $|\frac{1}{n} - \frac{1}{x}| \le \frac{1}{n^2}$ Prove $|\frac{1}{n} - \frac{1}{x}| \le \frac{1}{n^2}$ for all $n \le x \le n+1$, using the mean value theorem applied to $f(x) = \frac{1}{x}$ 
Immediately, I can recognize some components of the mean value theorem. $\frac{1}{n^2}$ likely comes from the slope of $f$ at $n$, and the  $|\frac{1}{n} - \frac{1}{x}|$ expression likely comes from the secant expression $\frac{\frac{1}{n} - \frac{1}{x}}{x}$. But I cannot figure out how they fit together in the end.
 [/INST] $n-x\leq1 $ therefore  $$|\frac{1}{n} - \frac{1}{x}| =|\frac{n-x}{nx}|\leq|\frac{1}{nx}|$$ i know $n \le x $ then $$\leq|\frac{1}{nx}|\leq\frac{1}{n^2}$$
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