Why is gravitational potential energy negative?

While on the Earth's surface, you need to put in energy to move upwards, due to the force of gravity from the Earth's mass acting on you - eg jumping upwards requires energy.
The force of gravity is strongest the closer you are to the source of it (eg the planet), and weaker the further you are from it (eg it is zero at an infinite distance)
At an infinite distance, there is no gravitational force acting on you. This means there is also no ability for you to be moved by the force, or in other words your potential energy must be zero.
However, as you move closer to Earth, your potential energy must decrease - the only way this is possible is for it to be negative.
Mathematically:
We know that Newton's law of gravitation is: F = - (GMm)/(r^2)
Minus sign shows it is an attractive force, ie you move opposite to the vector extending radially from the Earth to yourself in free space
And we know a potential associated with a force is: U = - Int[F.dx]
Hence V = -Int[-(GMm)/(r^2).dr]
= Int[(GMm)/(r^2) .dr]
= -GMm/r
At r = infinity, V = 0; at r<infinity, V<0