In deriving the expression for the exact propagator

$$G_c^{(2)}(x_1,x_2)=[p^2-m^2+\Pi(p)]^{-1}$$

for $\phi^4$ theory all books that i know use the following argument:

$$G_c^{(2)}(x_1,x_2)=G_0^{(2)}+G_0^{(2)}\Pi G_0^{(2)}+G_0^{(2)}\Pi G_0^{(2)}\Pi G_0^{(2)}+...$$

Here $\Pi$ is the sum of all irreducible diagrams.

Using Feynman diagrams to the lower order we can see that this is true, but what about the higher orders  ? Is there any formal proof (by induction or something else  ) that this true? I would appreciate links to articles or text books.

In deriving the expression for the exact propagator

$$G_c^{(2)}(x_1,x_2)=[p^2-m^2+\Pi(p)]^{-1}$$

for $\phi^4$ theory all books that i know use the following argument:

$$G_c^{(2)}(x_1,x_2)=G_0^{(2)}+G_0^{(2)}\Pi G_0^{(2)}+G_0^{(2)}\Pi G_0^{(2)}\Pi G_0^{(2)}+\ldots .$$

Here $\Pi$ is the sum of all irreducible diagrams.

Using Feynman diagrams to the lower order we can see that this is true, but what about the higher orders? Is there any formal proof (by induction or something else) that this true?