Ytukyg

I was reading a proof about the Lipschitz approximation of functions $uin W^{1, p}(mathbb{R}^n)$. There the author defines a set
$$E_{lambda}={xinmathbb{R}^n:M|nabla u|(x)leq lambda}, quad lambda>0, $$
where $M$ is the centered maximal function
$$
Mf(x)=sup_{r>0}frac{1}{|B(x, r)|}int_{B(x, r)}f(y)text{d}y.
$$

Then he shows that $u$ is $clambda$-Lipschitz a.e. in $E_{lambda}$. Then using McShane extension theorem one can find a $clambda$-Lipschitz function $v:mathbb{R}^nrightarrowmathbb{R}$ s.t. $v=u$ a.e. in $E_{lambda}$. Now we use truncation for the function $v$ by defining a new function $v_{lambda}=min{lambda, max{v, -lambda}}$, or
$$
v_{lambda}(x)=begin{cases} v(x) & |v(x)|leqlambda \ lambda & v(x)>lambda \ -lambda & v(x)<-lambda.
end{cases}
$$
Now comes the part I don't really understand. He claims that $v_{lambda}$ is $2clambda$-Lipschitz, but in my opinion $v_{lambda}$ is $clambda$-Lipschitz. Also he states that $v_{lambda}=u$ a.e. in the set $E_{lambda}$, but I'm not able to see that myself. Also, does it follow from this that the weak gradients also agree, i.e. $nabla v_{lambda}=nabla u$ a.e. in $E_{lambda}$?

Any help is appreciated!