I have been reading the text "Analysis on Fractals" of Jun Kigami. There is a theorem about fundamental system of neighborhoods of a point in a self similar set.

It is stated as follows

Let $\mathcal{L}=(K, S, \{F_i \}_{i\in S})$ be a self similar structure. For any $x\in K$ and any $m\geq 0$ define $$K_{m,x}=\bigcup_{w\in W; x\in K_w} $$ Then $\{K_{mx}\}_{m\geq 0}$ is a fundamental system of neighborhoods of $x$.

I will write a part of the proof, the part I think is wrong (or that I am not understanding well)

First he prove that

$$\displaystyle\max_{w\in W_m} \text{diam } K_w \rightarrow 0$$

when $m\rightarrow \infty$.

Finally he proves that every $K_{m,x}$ contains an open that in turn contains $x$:

Secondly, we show that $K_{m,x}$ is a neighborhood of $X$. Let $\{x_m\}_{m\geq 1}$ be a sequence in $K$ which converges to $x$ as $m\rightarrow\infty$. Choose $w^m\in \pi^{-1}(x_n)$ for any $m\geq 1$. Then there exists a sub sequence $\{w^{m_i}\}_{i\geq 1}$ that converges to some $w\in \Sigma$ as $i\rightarrow \infty$. Hence $x_{m_i}\in K_{m,x}$ for sufficiently large $i$.Therefore $K_{m,x}$ is a neighborhood of $x$.

I don't quite understand why because of the property he proved he concludes that $K_{m,x}$ is a neighborhood of $x$. I think that one has to prove that the sequence is eventually in $K_{m,x}$ instead of proving that there is a sub sequence that is eventually in $K_{m,x}$.

I tried to build a proof without using that argument, but I found noting.

Any comments?

Thanks!