So I was reading the Deformation Theorem of Chapter 8.5 of "Partial Differential Equations" by Evans, and I'm confused about one step of the proof.
First, some definitions. $H$ is a real Hilbert space with norm $||\cdot||$ and inner product $(\cdot,\cdot)$. Also, $I:H \to \mathbb{R}$ is a nonlinear functional.
DEFINITION. We say $I$ is differentiable at $u \in H$ if there exists $v \in H$ such that $$I[w]=I[u]+(v,w-u)+o(||w-u||) \quad (w \in H)$$ The element $v$, if it exists, is unique. We then write $I'[u]=v$.
DEFINITION. We say $I$ belongs to $C^1(H;\mathbb{R})$ if $I'[u]$ exists for each $u \in H$ and the mapping $I':H \to H$ is continuous.
We also write $\mathcal{C}$ to represent the collection of functions $I \in C^1(H;\mathbb{R})$ such that $I':H \to H$ is Lipschitz continuous on bounded subsets of $H$.
Now, suppose $I \in \mathcal{C}$, $0<\delta<\epsilon$, and define the sets
\begin{equation*} \begin{aligned} & A:=\{u \in H \mid I[u] \leq c-\epsilon \text { or } I[u] \geq c+\epsilon\}, \\ & B:=\{u \in H \mid c-\delta \leq I[u] \leq c+\delta\} . \end{aligned} \end{equation*}
The claim is
Since $I'$ is bounded on bounded sets, we verify that the mapping $u \mapsto dist(u,A) +dist(u,B)$ is bounded below by a positive constant on each bounded subset of $H$.
Here's the proof that I came up with:
Suppose by contradiction that there exists $u_o \in H$ such that $dist(u_0,A)+dist(u_0,B)=0$. Thus, $dist(u_0,A)=0$ and $dist(u_0,B)=0$, which is equivalent to $u_0 \in \bar{A} \cap \bar{B}$ (where the bar represents the closure).
Now, let $v \in \bar{B}$. Then there exists a sequence $v_n \in A$ such that $v_n \to v$ in H. Since $I$ is continuous, $I[v_n] \to I[v]$, and thus $c-\delta \leq I[v] \leq c+ \delta$, so $v \in B$, thus $B=\bar{B}$ is closed.
By a similar reasoning, we get that $A=\bar{A}$ is closed. so $u_0 \in \bar{A} \cap \bar{B}= A \cap B = \emptyset$, a contradiction.
So I have some doubts about this "proof", since I never used the hypothesis that $I'$ is bounded on bounded sets, although I can't see the error in it. Could someone point out my mistake? Is it because $u_0$ in the conditions above may not exist? I am also aware of a correct way to prove this, as can be seen in On the proof of deformation lemma "boundedness", but I'd like to know if my reasoning can be salvaged.