There is no absolute proof until we go out there and measure it directly, but the current theory is well supported by multiple lines of evidence.

First is the cosmological redshift, which is *not* the same as the Doppler shift. While the Doppler shift reflects the relative motion between the object and the observer (and can therefore be a redshift or a blueshift), the cosmological or Hubble redshift is a direct result of the expansion of space itself. As the universe expands, so do the light waves traveling through it, and the amount of wave expansion (the redshift) increases with time after the light is first emitted. Therefore, the light from distant objects, which has been traveling for billions of years, appears "stretched" to longer and longer wavelengths.

http://www.astro.virginia.edu/~jh8h/glossary/redshift.htm

This raises a question: How do we know that highly redshifted objects are really as far away as we think they are? We can't measure the distance, and maybe there's another explanation for such high redshifts. However, there is additional evidence to support the expansion theory as an explanation for the redshift.

First, the cosmic background radiation has a spectrum which matches what would be expected from the "first light" of the universe after having become expanded to radio wavelengths over 13+ billion years.

http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation

This may be a bit self-defining, but the second line of evidence is not. Certain types of stars behave in very specific ways related to their absolute magnitude (brightness), which makes them useful as "standard candles" to determine very large distances. Given an example close enough to Earth for its distance to be measured accurately, the distances to similar objects can be derived directly from their apparent magnitudes.

http://universe-review.ca/R02-07-candle.htm

The standard candle that supports expansion is the type Ia supernova. They are the result of a white dwarf star accumulating mass from a nearby companion star until it approahes the ultimate "critical mass" and undergoes runaway fusion. Since type Ia supernovae all have the same mass and detonate in the same way, they all have about the same brightness, and the apparent magnitude is directly related to the distance to the supernova.

http://universe-review.ca/R02-07-candle.htm#type1a
http://en.wikipedia.org/wiki/Type_Ia_supernova

Until one or more of these lines of evidence is disproven, the expansion of the universe will continue to be accepted, if not as "fact," then at least "consistent with everything else we know."

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