On the proof of deformation lemma "boundedness" Book- Evans partial differential equation. In the proof of deformation lemma how to say that  $V(u)=-g(u)h(\lVert I'(u)\rVert)I'(u)$ is bounded. And how to say that the mapping $u \to \operatorname{dist}(u,A)+\operatorname{dist}(u,B)$ is bounded.
 A: For the boundedness of $V$: note that $g$ is bounded and that, by the definition of $h$, we have $\left\Vert h(\Vert I'(u)\Vert)\, I'(u)\right\Vert=\Vert I'(u)\Vert$ if $\Vert I'(u)\Vert\leq 1$ and $\left\Vert h(\Vert I'(u)\Vert)\, I'(u)\right\Vert=1$ if $\Vert I'(u)\Vert\geq 1$.
For the second question, what is claimed in Evans' book is that the function $u\mapsto d(u,A)+d(u,B)$ is bounded below on bounded sets, i.e. for any bounded set $U$, one can find $\delta_U>0$ such that $d(u,A)+d(u,B)\geq \delta_U$ on $U$. So let us prove that.
Let $U$ be any bounded set in $H$, and choose $R$ such that $U\subset B(0,R)$. Towards a contradiction, assume that one can find a sequence $(u_n)\subset U$ such that $d(u_n,A)+d(u_n,B)\to 0$. For each $n\in\mathbb N$, choose $a_n\in A$ and $b_n\in B$ such that $\Vert u_n-a_n\Vert<d(u_n,A)+2^{-n}$ and $\Vert u_n-b_n\Vert\leq d(u_n,B)+2^{-n}$. Then $\Vert u_n-a_n\Vert+\Vert u_n-b_n\Vert\to 0$, so $\Vert a_n-b_n\Vert\to 0$ by the triangle inequality. Moreover, all points $a_n,b_n$ lie in the ball $B(0,R+1)$ since $u_n\in B(0,R)$ for all $n$. Since $I'$ is bounded on bounded sets, it follows (by the mean value theorem) that $\vert I(a_n)-I(b_n)\vert\to 0$. But this is a contradiction because $a_n\in A$ and $b_n\in B$, so that $\vert I(b_n)-I(a_n)\vert\geq \varepsilon-\delta$ for all $n$. 
