Proving that the universe is infinite

Let`s imagine a library from one cylindrical chamber and suppose it`s a finite universe; every book must contain a part from the existing information and images so that all the data is contained; also the information and the images of them – of their covers, of their files and of the ensemble cover-pages/ exterior- interior- must be contained.
From here we distinguish two possible cases: the data about the books can be divided and shared between them or we have another book which contains the data and the images about the others.
In any of the cases, there still remains data not included: case 1.the image of the book containing solely the information about others and 2.the images of the ``new books``, the old books containing now more information, which asserts a new form and more space.
If we try to incorporate them in: a bigger book, there still remains free data about that book or in the same volumes, adding files, we still have this second generation of containers to be added to The Library.
So, whatever we do, we always have the newest forms/containers not contained. The essence: if we try to have all the information we need a place to put it and for the information about that place we need another space and so on, ad infinitum.
As an universe needs to contain everything, it needs to contain his actual form and, as he can`t do this as a limited universe or as a ``possibly infinite`` universe (because in order to contain himself it needs to update so the ``newest`` form is never contained even if grows progressively and becomes a ``possibly infinite`` universe*) we can have only an ``actually infinite`` universe.
Finally, I can`t make suppositions in the domain of physics but I can say only that it will never prove the universe if limited: it will prove only that a part of our universe is limited.
And, why not, if the universe is infinite, then there can`t be more universes or multiverses, because they would juxtapose. There can be only places outside the ``bubble`` of matter we live in/ places that don`t respect the laws of physics, if the ``bubble`` of matter is actually infinite.

Note: here are the definitions of the universe:

``the universe: all of space and everything in it`` Merriam Webster;

``The Universe is all of time and space and its contents`` Wikipedia;

`` The sum of everything that exists in the cosmos, including time and space itself`` Wiktionary.

*The assumption that no finite universe ever reaches infinity doesn`t assume it`s still finite, because it grows continuously and progressively and can`t be measured.