Scientists don't like to use the word "proof" because everything in science is subject to revision as new data comes in. But the case for human-caused global warming is about as strong as it gets.
1. World surface temperatures are getting warmer, and this trend has accelerated since the mid 1970's. Almost no scientist in the 21st century has disputed this basic fact, even among the most diehard GW skeptics. Here is the data from NASA / GISS:
http://data.giss.nasa.gov/gistemp/tabledata/GLB.Ts+dSST.txt
... and from the UK's Hadley Centre:
http://hadobs.metoffice.com/hadcrut3/
As I said, even GW skeptics accept that it's getting warmer; the major dispute is what's causing it: human beings, through increased greenhouse gases in the air? Or natural causes, like the Sun? The dispute is more political than scientific, though, because the scientific case for increased greenhouse effect is rock solid.
If the Sun is causing the current warmth, then we're getting more energy, and the whole atmosphere should be getting warmer. If it's greenhouse, then we're getting the same amount of energy, but it's being distributed differently: more heat is trapped at the surface, and less heat is escaping to the stratosphere. So if it's the Sun, the stratosphere should be warming, but if it's greenhouse, the stratosphere should be cooling.
In fact, the stratosphere has been on a long-term cooling trend ever since we've been keeping radiosonde balloon records in the 1950's. Here's the data:
http://hadobs.metoffice.com/hadat/images/update_images/global_upper_air.png
http://hadobs.metoffice.com/hadat/hadat2/hadat2_monthly_global_mean.txt
http://cdiac.ornl.gov/trends/temp/sterin/sterin.html
2. If it's the Sun, we're getting more energy during the day, and daytime temperatures should be rising fastest. But if it's greenhouse, we're losing less heat at night, and nighttime temperatures should be rising fastest. So if it's the sun, the difference between day and night temperatures should be increasing, but if it's greenhouse, the day-night difference should be decreasing.
In fact, the daily temperature range has been decreasing throughout the 20th century. Here's the science:
http://ams.allenpress.com/perlserv/?request=get-abstract&doi=10.1175%2F1520-0450(1984)023%3C1489:DDTRIT%3E2.0.CO%3B2
http://ams.allenpress.com/perlserv/?request=get-abstract&doi=10.1175%2F1520-0477(1993)074%3C1007%3AANPORG%3E2.0.CO%3B2
http://www.bom.gov.au/bmrc/clfor/cfstaff/jma/2004GL019998.pdf
3. Total solar irradiance has been measured by satellite since 1978, and during that time it has shown the normal 11-year cycle, but no long-term trend. Here's the data:
http://www.ngdc.noaa.gov/stp/SOLAR/solarda3.html
4. Scientists have looked closely at the solar hypothesis and have strongly refuted it. Here's the peer-reviewed science:
http://www.pubs.royalsoc.ac.uk/media/proceedings_a/rspa20071880.pdf
http://www.mpa-garching.mpg.de/mpa/publications/preprints/pp2006/MPA2001.pdf
5. CO2 levels in the air were stable for 10,000 years prior to the industrial revolution, at about 280 parts per million by volume (ppmv). Since 1800, CO2 levels have risen 38%, to 384 ppmv, with no end in sight. Here's the modern data...
http://www.esrl.noaa.gov/gmd/ccgg/trends/
... and the ice core data ...
http://www.ncdc.noaa.gov/paleo/icecore/antarctica/law/law.html
http://www.ncdc.noaa.gov/paleo/icecore/antarctica/domec/domec_epica_data.html
http://www.ncdc.noaa.gov/paleo/icecore/antarctica/vostok/vostok_data.html
... and a graph showing how it fits together:
http://www.columbusnavigation.com/co2.html
6. We know that the excess CO2 in the air is caused by burning of fossil fuels, for two reasons. First, because the sharp rise in atmospheric CO2 started exactly when humans began burning coal in large quantities (see the graph linked above); and second, because when we do isotopic analysis of the CO2 we find increasing amounts of "old" carbon combined with "young" oxygen. Here are the peer-reviewed papers:
http://adsabs.harvard.edu/abs/1984JGR....8911731S
http://www.ingentaconnect.com/content/mksg/teb/1999/00000051/00000002/art00005
http://www.sciencemag.org/cgi/content/abstract/256/5053/74
So what's left to prove?