This question has been being asked for years and answered for years. Still it remains intact in the hearts of many. Also there are two categories of such people who question the existence of God, the first category consists of the people who will accept the existence of God if proven to them. Whereas the second category consists of people who remain rigid even when logical statements are presented before them.
Now many people try to prove the existence of God by stating that a creation needs a creator thus there must be a creator of the universe who possess higher energy and position.
This argument is weak as if we accept this logic then we have to accept that even God is created by someone. Thus this argument will not stand strong.
So to prove existence of God, Shankaracharya gives a very solid logical argument.
Every person in this world desires to attain happiness, there is not a single person who does not ever wishes to be distressed. Thus we can say that everyone is looking for Anand (Sanskrit name for happiness).
Now the second thing desirable by every single person is immortality, no one wants to die yet everyone dies, though in reality the soul is immortal it is only the body which perishes, but if we accept this then there arises a counter argument stating that if everyone desires immortality then why do people commit suicide, so to understand this we have to remind ourselves that everyone desires ‘Anand’ and when people are in such a state of distress where they don’t see any other option of attaining ‘Anand’ they commit suicide, thus this counter argument does not make the statement made futile. Thus we can say that everyone also desires ‘Sat’ (eternal existence).
Now everyone, be it any person also desires to know everything, to have the absolute perfect knowledge about everything, to possess every skill/ability. Thus we can say that the third desirable thing by everyone is ‘Cit/Chit’ (knowledge).
Thus we can now say that everyone desires ‘Sat’, ‘Cit’, and ‘Anand’ for their entire life.
And we can desire for something which exists in the world, thus we have to accept that ‘Sat’ ‘Cit’ ‘Anand’ exist, thus Bhagvan who is Sadcidanand exists.