Compact set of measure zero and sequence of Harmonic Functions with nice properties. I was studying John B. Garnett's book Bounded Analytic Functions, and then I decided to try the following problem:

Let $E \subset \mathbb{R}$ be a compact set, with $|E|=0$. Prove that there exists a sequence of Harmonic functions $V_n(z)$ defined on the upper half space $\mathcal{H}$ such that
(i) $V_n \ge 0 $
(ii) $\lim_{\mathcal{H}\ni z \rightarrow t} V_n(z) = + \infty, \;\forall t \in E$
(iii) $\lim_{n \rightarrow \infty} V_n(z) = 0 \;\;\forall z \in \mathcal{H}$

At first, I thought this would be a routine exercise, as the texts builds a brief but fair theory about Poisson Integrals.
So, my first idea was to construct a sequence of functions $\{f_n\}$ such that they satisfied the following properties:
(a) $f_{n-1} \ge f_n \ge 0$, with $f_n \rightarrow 0$ a.e.
(b) $\int_{\mathbb{R}} f_n dx < + \infty$
(c) $f|_{E} = +\infty$
and then define $V_n(z) := P_y \star f_n (x)$, if $z = x + i y$.
But, up to this moment, I could not find a proper sequence of functions, as for everyone I try, they fail to accomplish condition (ii).
Any ideas how to fix that?
 A: Today I sat down with a friend to try to solve this problem, and we think that we may have gotten a solution: 
First, let's prove some properties about convolutions of the type $P_y * \chi_I (x)$, $I$ a bounded interval: If $I = [a,b]$, $x \in I$, $0\le y \le \frac{b-a}{2}$, then 
$$ P_y * \chi_I(x) = \int_{x-b}^{x-a} P_y(t) dt \ge \int_{0}^{(b-a)/2} P_y(t) dt \ge \int _{0}^y P_y(t) dt = \frac{\pi}{4} $$
As $|E|=0$, we may find a collection of finitely many bounded disjoint intervals $\{I_j^n\}$ such that $\sum_j |I_j^n| \le n^{-2}$ and $E \subseteq \bigcup_j I_j^n$. Let then $g_n = \sqrt{n} \sum_j \chi_{I_j^n} $. 
We are now going to define $f = \sum_{n \ge 1} g_n$. We see immediately that
$$ \int f(x) dx \le  \sum_{n\ge 1} n^{-3/2} < \infty $$
But, if $y \le \min_j \{|I_j^n|\}$, then, for $x \in  I_j^n$, we've that 
$$ P_y * f (x) \ge \sqrt{n} P_y * \chi_{I_j^n} (x) \ge \sqrt{n} \frac{\pi}{4} $$
And it follows that condition (ii) holds for the function $V (z) := P_y * f (x)$ 
Condition (i) for $V$ is now direct from the definition. Defining $V_k = \frac{V}{k}$, we get our required sequence of Harmonic Functions. 
