I have very little knowledge on inverse limits, in fact I only started reading about it when I came across it on this particular proof for a theorem on the general criterion of the existence of roots of 1-Lipschitz functions. I hope someone could help me understand this part of the proof:
Given:
$\bullet f: \mathbb{Z}_p \rightarrow \mathbb{Z}_p \text{ f is 1-Lipschitz and continuous }\\ \bullet f(x)\equiv 0\pmod{p^k} \text{ has a solution } \forall k >0 \\ \bullet M_k \subset \{0,1,2,...,p^k-1\} \text{ be the set of all solutions of the congruence above}\\ \bullet h_i \in M_i \\ \bullet \mathbb{M} \text{ is the set of all possible sequences } \{h_1, h_2, h_3, ..., h_k,... \} \\ \bullet f(b_i) \equiv f(a_k) \equiv 0\pmod{p^i}, \text{ such that } b_i \in M_i \text{ and } a_k\in M_k \\ \bullet \varprojlim M_k = \mathbb{M}$
And this is the excerpt from the proof which I want to understand on how they came up with the conclusion
$$ \text{Since } f \text{ is continuous }\\ \text{then } f(a)=0, \forall a\in \mathbb{M} $$
I hope someone could explain to me how that happened. Thank you!