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Smashing atoms in an accelerator proves that atoms are made of smaller particles which themselves are made of smaller particles. An accelerator is needed to reach the minimum energy level required to do this though, atoms don't just simply disintegrate themselves, unless in a fission nuclear reactor.

Einstein proved that atoms exist when he studied Brownian motion, which is what happens when you put grains of pollen in water. The move around without any apparent objected causing it. Einstein proved that the grains moved because atoms were causing that motion.

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