No doubt you can prove this.
Simple.
I live on Earth, and there is life here.
I cannot visit other planets, therefore there is no life there.
q.e.d.
Hey, I want to believe! Can you share ANY evidence that there is life in other places?
As far as what we know, space beyond the reach of out interstellar probes could be an elaborate projection. As silly as that sounds, it is just as possible as any other construct. The possibility of life on other worlds is speculation without evidence.
There are numerous reasons why the number of planets in our galaxy capable of sustaining carbon-based life is much smaller than the media and grant-seeking astronomers imply in their articles. In addition to the obvious life-hospitable factors, such as a temperature suitable for liquid water and a suitably-sized sun, there are many more factors required for a planet to be fit for human-like intelligent life:
Suitable gravity for atmosphere, with appropriate pressure, to retain oxygen and water vapor but not hydrogen; suitable axis rotation rate; appropriate tilt of planet axis for seasons; suitable orbital distance from star to prevent tidal locking; magnetic field from planet molten core to shield atmosphere from cosmic radiation; sufficient planetary plate tectonics for crustal mineral recycling; suitable distribution and shape of continents to sustain deep ocean currents; sufficient actinide radioisotopes in planet interior for decay heat; suitably-sized moon to create needed tides and stabilize planet rotation; giant outer gas planets to prevent large asteroids and comets from frequent collisions with planet; suburban galaxy location providing sufficient distance from recent supernovas; far enough away from gravitational disturbances by higher density of stars in inner area of galaxy; only minor orbital variations caused by small chaotic gravitational interactions with nearby planets.
More limiting factors on suitable plaents will likely to be discovered as we learn more about our galaxy. The estimated probable number of life-hospitable planets is likely very close to the experimentally known value of one.
It is an inductive proof.
Coppedge showed that the chances of Amino Acids forming from Polypeptides was 1 in 100 trillion trillion. There are hundreds of thousands of these required for the simplest one-celled life. So the chances of life forming by chance would have required thousands of trillions of years just to come within the 10^-50 probability range of possible.
Flying saucers are fascinating. But they are no more evidence of extraterrestrial craft than the UFO flap occurring in the 1890s recording their speed at THE INCREDIBLE RATE OF 200 MPH, PROOF of extraterrestrial involvement at the time.