The great Upanishads have clearly stated that the Ultimate Reality (UR), Brahman or “God”, is That which the eyes cannot see but because of which the eyes can see. It is That which no sense can perceive but because of which the senses can perceive. Mind cannot feel That nor intellect comprehend It; but mind has the ability to feel and intellect the power to comprehend because of That. Brahman is the ultimate ground or support for everything there ever is to be seen; It is also what gives the seer the power to see and know. The battery inside a flashlight gives it the power to see everything around, but that flashlight does not see the battery itself. Similarly, Brahman being the very ground on which the seer and the seen exist, efforts to “see” or “know” It must fail. Attempts to “prove it” are also likely fool’s errands. If a proof is to be found to establish Ultimate Reality, then that proof must contain a truth superior to the Ultimate Reality. This is contrary to the definition of Ultimate Reality.
There is, however, no need to prove Brahman. Brahman is said to be self-evident or “self-luminous”. It is as self-evident as your existence is to you. You do not need anyone else to know you or prove your existence to know that you exist. You and Brahman are both self-evident and qualify as the Ultimate Reality. Therefore, we have the well known Vedantic declaration: “You are That (Brahman)”. You are, UR, the Ultimate Reality.
So, if you find yourself searching for God, remember the saying “what you are searching for, you are searching with!”
A Related Question: If Brahman cannot be known, what is the purpose of the body of knowledge known as Vedanta? See the separate post on “Ignorance (avidya) and Its Elimination Through Knowledge”. For an answer, please see the separate post on the purpose of spiritual knowledge.