Math has proven that the universe definitely had a start. According to two cosmologists, Audrey Mithani and Alexander Vilenkin from Tufts University in Massachusetts, they came out with a new mathematical paper that considers the mathematics of eternity. They took a close look at the concept of a universe that has no beginning or end.
The main arguments suggest that the universe is eternally old, without a Big Bang, but the first description of it is an eternal inflation model, which different parts of the universe expand and contract at different rates.
The other idea of an emergent universe, one which exists as a kind of seed for eternity and then suddenly expands into life.
The idea of an eternal universes can only allow certain types of universe expansion to occur, and when they do the current inflation models that have been suggested have to have a beginning to it. The math is pretty complex, but you can read their paper at the link at the endof the article. Here's how they sum it up:
"Although inflation may be eternal in the future, it cannot be extended indefinitely to the past."
"A simple emergent universe model...cannot escape quantum collapse."