understand a part of the proof of Rademacher's theorem in the book of L.C. Evans and R.F.Gariepy . Here it is given that 

$$\limsup_{t\to0}\dfrac{f(x+tv)-f(x)}{t}= \lim_{k\to\infty}\sup_{{0<|t|<1/k} , t\in\mathbb Q}\dfrac{f(x+tv)-f(x)}{t}$$

where $f:\mathbb R^n\to \mathbb R^m$ is continuous on $\mathbb R^n$, $v\in\mathbb R^n$ with $|v|$=1.
Please someone give some hints why this equality holds.
 A: Let $\ell=\limsup_{t\to 0}\frac{f(x+tv)-f(x)}{t}$. Assume that $\ell\ge 0$. Then there exists $t_n\to 0$ such that $$\ell=\lim_{n\to \infty}\frac{f(x+t_n v)-f(x)}{t_n}.$$
Fix $k$. Since $t_n\to 0$ there is $n_k$ such that $0<|t_n|\le \frac1{2k}$ for all $n\ge n_k$. By the density of the rationals, for each $t_n$ there exists $s_n\in\mathbb{Q}$, $s_n\ne 0$ such $|s_n-t_n|<\frac{|t_n|}{2k}<\frac1{2k}$. Since $f$ is Lipschitz continuous and $|s_n/t_n-1|<\frac{1}{2k}$,
$$\left\vert\frac{f(x+t_n v)-f(x)}{t_n}\right\vert\le \left\vert\frac{f(x+t_n v)-f(x+s_n v)}{t_n}\right\vert+\frac{|s_n|}{|t_n|}\left\vert\frac{f(x+s_n v)-f(x)}{s_n}\right\vert
\\\le L \frac{|t_n-s_n|}{|t_n|}+\left(1+\frac1k\right)\sup_{0<|s|<\frac1k,\, s\in \mathbb{Q}} \left\vert\frac{f(x+s v)-f(x)}{s}\right\vert
\\\le L \frac{1}{k}+\left(1+\frac1k\right)\sup_{0<|s|<\frac1k,\, s\in \mathbb{Q}} \left\vert\frac{f(x+s v)-f(x)}{s}\right\vert.$$
Letting $n\to\infty$ gives
$$\ell \le L \frac{1}{k}+\left(1+\frac1k\right)\sup_{0<|s|<\frac1k,\, s\in \mathbb{Q}} \left\vert\frac{f(x+s v)-f(x)}{s}\right\vert.$$
Then letting $k\to\infty$ gives
$$\ell\le \lim_{k\to\infty}\sup_{0<|s|<\frac1k,\, s\in \mathbb{Q}} \left\vert\frac{f(x+s v)-f(x)}{s}\right\vert.$$
The other inequality is simpler. Do you need more details?
