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1. Each person of the Trinity has all of God's attributes.


2. Being omnipresent (not being bounded by any location, to be present everywhere) is one of God's attributes.


3. So, each person of the Trinity has the attribute of being omnipresent.


4. If one person is not another, then there is something else besides this person.


5. If there is something else besides this person, then this person is bounded by this something else.


6. Whatever is bounded cannot be omnipresent. Hence a contradiction.

You cannot use words like “each” to describe the “triune god”. The word “each” implies separate parts of a whole, but the divine substance does not have “parts”, and that which it is cannot be seperated. One “attribute” of the divine substance is that it exists as three consubstantial persons, so by it’s own definition, the only “actual infinity” to ever exist is three. This can easily be proven mathematically because 3(∞)= ∞... of course so does 4(∞), 5(∞), ... ∞. But Nicea chose three, so it’s three.

This is a partial answer hopefully getting to another question that deals with the apparent multiplicity in God that the Trinity represents.

Not all religions associated with Christianity need to assert the idea of the Trinity. For example Unitarian Universalists with roots in liberal Christianity do not appear to do so:

Unitarian Universalism (UU) is a liberal religion characterized by a "free and responsible search for truth and meaning". Unitarian Universalists assert no creed, but instead are unified by their shared search for spiritual growth. As such, their congregations include many atheists, agnostics, and theists within their membership. The roots of Unitarian Universalism lie in liberal Christianity, specifically Unitarianism and universalism. Unitarian Universalists state that from these traditions comes a deep regard for intellectual freedom and inclusive love. Congregations and members seek inspiration and derive insight from all major world religions.

What is important in the argument is to identify more precisely who holds these beliefs.

Once one knows who holds these beliefs, one needs to find representatives from those groups, preferably the most respected representatives, and quote their exact views on the Trinity paying attention to their arguments.

Then one can attempt to use those arguments to try to reach a contradiction.

Without getting those quotes and identifying whom one is opposing, one is setting up a straw man. This is how Wikipedia describes a straw man:

A straw man is a common form of argument and is an informal fallacy based on giving the impression of refuting an opponent's argument, while actually refuting an argument that was not presented by that opponent. One who engages in this fallacy is said to be "attacking a straw man."

The issue of having multiplicity in unity in general should raise philosophical concerns which are worth exploring even with particular examples such as the three Persons in one God in the concept of the Christian Trinity. One simply needs to be precise about what people are claiming when they speak of these particular multiplicities.

Reference

Wikipedia, "Straw man argument"

Wikipedia, "Unitarian Universalism"

Your proof is valid.

Here's an analogy:

Assume the observable universe is infinite.

Define infinite to mean contains everything.

If something is outside the observable universe, the universe does not
contain everything; it is not infinite.

Some Christians don't define infinite that way (God is creator, not creation¹). They would reject the soundness of your proof. Specifically they would reject 6: bounded as you defined it implying not infinite. God is infinite in knowledge. God is infinite in power. But God is not everything. For example Satan is not God. For God is good. And Satan is not.

Also some Christians view the Trinity as a mystery. They would dismiss any attempt to understand the Trinity beyond what they believe is explained in the Bible.

^{1. "Christian teachings of the immanence and involvement of God and his love for humanity exclude the belief that God is of the same substance as the created universe" -- Wikipedia, "God in Christianity"}

"Has infinite extent" is not equivalent to "no part of something is not part of it."

Trivial example: there are points in a 3d surface that are not part of an infinite plane.

Better example: We define the region Z to be everything in 3d space outside of the unit sphere. There are points (in fact an infinite number of them but I digress) inside the unit sphere that are not part of region Z. Z has infinite continuous volume. I don't remember the exact notation for levels of infinity but it doesn't matter here. ∞ - ∞ is not well defined.

God's attributes are not contextless: God isn't infinitely everything. God is not infinitely sinful, or infinitely ignorant, or infinitely cruel. He is infinitely wise, and infinitely powerful, and even infinitely good, although we may find it difficult to see his infinite goodness when he allows and even works through bad things, though that is a limitation we have, not something that prohibits God from being truly infinitely good.

There is neither an infinite number of persons in the godhead, nor does each person infinitely exist to the exclusion of the other persons of the godhead. Neither of those are contexts where the infiniteness of God applies.