It is amazing that debates on Big Bang theory are still out there
It has been proven already
Anyway,There are many proofs.
Cosmic Microwave background radiation:
Perhaps the most conclusive (and certainly among the most carefully examined) piece of evidence for the Big Bang is the existence of an isotropic radiation bath that permeates the entire Universe known as the "cosmic microwave background" (CMB). The word "isotropic" means the same in all directions; the degree of anisotropy of the CMB is about one part in a thousand. In 1965, two young radio astronomers, Arno Penzias and Robert Wilson, almost accidentally discovered the CMB using a small, well-calibrated horn antenna. It was soon determined that the radiation was diffuse, emanated unifromly from all directions in the sky, and had a temperature of approximately 2.7 Kelvin (ie 2.7 degrees above absolute zero). Initially, they could find no satisfactory explanation for their observations, and considered the possibility that their signal may have been due to some undetermined systematic noise. They even considered the possibility that it was due to "a white dielectric substance" (ie pigeon droppings) in their horn!
However, it soon came to their attention through Robert Dicke and Jim Peebles of Princeton that this background radiation had in fact been predicted years earlier by George Gamow as a relic of the evolution of the early Universe. This background of microwaves was in fact the cooled remnant of the primeval fireball - an echo of the Big Bang.
If the universe was once very hot and dense, the photons and baryons would have formed a plasma, ie a gas of ionized matter coupled to the radiation through the constant scattering of photons off ions and electrons. As the universe expanded and cooled there came a point when the radiation (photons) decoupled from the matter - this happened about a few hundred thousand years after the Big Bang. That radiation cooled and is now at 2.7 Kelvin. The fact that the spectrum (see figure) of the radiation is almost exactly that of a "black body" (a physicists way of describing a perfect radiator) implies that it could not have had its origin through any prosaic means. This has led to the death of the steady state theory for example. In fact the CMB spectrum is a black body to better than 1% accuracy over more than a factor of 1000 in wavelength. This is a much more accurate black body than any we can make in the laboratory!
By the early 1970's it became clear that the CMB sky is hotter in one direction and cooler in the opposite direction, with the temperature difference being a few mK (or about 0.1% of the overall temperature). The pattern of this temperature variation on the sky is known as a "dipole", and is exactly what is expected if we are moving through the background radiation at high speed in the direction of the hot part. The inference is that our entire local group of galaxies is moving in a particular direction at about 600 km/s. In the direction we are moving the wavelengths of the radiation are squashed together (a blue-shift), making the sky appear hotter there, while in the opposite direction the wavelengths are stretched out (redshift), making the sky appear colder there. When this dipole pattern, due to our motion, is removed, the CMB sky appears incredibly isotropic. Further investigations, including more recent ones by the COBE satellite (eg Smoot et. al.), confirmed the virtual isotropy of the CMB to better than one part in ten-thousand.
Einstein predicted the expansion of the universe before Hubble but he dismissed the idea of an expanding universe as Big Bang was not laid out then
Besides this,Hubble proved with his telescope that galaxies are getting farther away.