my question is how do you prove his existence?
Swami krishnananda logically proves (mostly from Advaitic perspective) God/Brahman exists in his book Lessons on the Upanishads.
Swamiji proves there is something which doesn't change in everything and says that is nothing but Brahman (ultimate Reality).
We begin to feel there must be something above this world. This was
what the great poets and the sages of the Vedas felt. Everything seems
to be transitory, moving, and in a state of flux. There is change in
nature, change in human history, change in our own mental and
biological constitution, change in even the solar system, the
astronomical setup of things. Everything is changing. The perception
of change is something very important for us to consider. How do we
know that things are changing, that things are moving or are
transitory? There is a logical peculiarity, a significance and a
subtlety at the back of this ability on our part to perceive change
and transition in things. A thing that changes cannot perceive change
by itself. Change cannot know change. Only that which does not
change can know that there is change.
This is a very important point at the rock bottom of our thinking that
we have to recognise. If everything is changing, who is it that is
telling us that everything is changing? Are we also changing with the
things that change? If that is the case, how do we come to know that
all things are changing? Logical analysis of this peculiar analytical
circumstance tells us that there is something in us which does not
change; otherwise, we would not know that things are changing.
Now, if oneself – this person or that person – seems to be obliged to
recognise something in one's own self that does not seem to be
changing because one perceives change in general, we also have to be
charitable enough to accept that everyone in the world has this
something which does not change. I have something in me which does not
change, and you also have something in you that does not change. If
this is the case, it seems to be everywhere. It does not mean that
this unchanging so-called thing is only in one person, as all persons
have an equal prerogative to conclude that something unchanging seems
to be there, speaking in a language which is not subject to connection
with changeable objects.
The Veda Samhitas to which I have made reference – which are the
outpourings of spiritual seekers, sages and masters of advanced
religious thought and spiritual perfection – felt the presence
everywhere of something that does not change. All things seem to be
embedded with something that cannot change. This is due to a logical
conclusion to which we are led – namely, that the perception of change
would not be possible if everything, including oneself, including even
the perceiver of change, also changes. Therefore, transitoriness
implies a non-transitory background of things.
The whole universe of perception, the entire creation, may be said to
be involved basically, at the root, in something which cannot be said
to change. This is an adorable and most praiseworthy conclusion, and
anything that is adorable is a worshipful something. These masters of
the Vedas Samhitas, therefore, recognised a divinity in all things.
There is a god behind every phenomenon, which is another way of saying
there is an imperishable background behind every perishable
phenomenon. The sun rises in the east, the sun sets in the west;
clouds gather, pour rain and then go; seasons change; something comes,
something goes; we are born, we become old and we also go. Everything
is changing, everywhere, even in the vast universe of astronomical
That which doesn't change (underlying reality) is called by various names such as Purusha, Brahman, Atman (which is same as Brahman as per Advaita), consciousness etc. You can read complete introduction to understand it in more detail.
What logically said by Swamiji "that which doesn't change is present in every being, every phenomenon" is present in Vedas also. For example, Sri Rudram says this ultimate reality (as Rudra) is present in everything . See this answer. Very first verse of Isha Upanishad states:
ईशावास्यमिदं सर्वं यत्किञ्च जगत्यां जगत् ।
‘Whatever there is
changeful in this ephemeral world, all that must be enveloped by the
Book God exists by Swami Sivananda might be also useful. Swami Sivananda explains logically by answering various questions of an atheist/agnostic and also quoting direct verses from scriptures.