Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 transitive v. To establish the truth or validity of by presentation of argument or evidence.
 transitive v. Law To establish the authenticity of (a will).
 transitive v. To determine the quality of by testing; try out.
 transitive v. Mathematics To demonstrate the validity of (a hypothesis or proposition).
 transitive v. Mathematics To verify (the result of a calculation).
 transitive v. Printing To make a sample impression of (type).
 transitive v. Archaic To find out or learn (something) through experience.
 intransitive v. To be shown to be such; turn out: a theory that proved impractical in practice.
 prove out To turn out well; succeed.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 v. To demonstrate that something is true or viable; to give proof for.
 v. To turn out; to manifest.
 v. To turn out to be.
 v. To put to the test, to make trial of.
 v. To experience
 v. Simple past of proove.
from the GNU version of the Collaborative International Dictionary of English
 transitive v. To try or to ascertain by an experiment, or by a test or standard; to test
 transitive v. To evince, establish, or ascertain, as truth, reality, or fact, by argument, testimony, or other evidence.
 transitive v. To ascertain or establish the genuineness or validity of; to verify.
 transitive v. To gain experience of the good or evil of; to know by trial; to experience; to suffer.
 transitive v. To test, evince, ascertain, or verify, as the correctness of any operation or result; thus, in subtraction, if the difference between two numbers, added to the lesser number, makes a sum equal to the greater, the correctness of the subtraction is proved.
 transitive v. To take a trial impression of; to take a proof of.
 intransitive v. To make trial; to essay.
 intransitive v. To be found by experience, trial, or result; to turn out to be
 intransitive v. To succeed; to turn out as expected.
from The Century Dictionary and Cyclopedia
 To try by experiment, or by a test or standard; test; make trial of; put to the test: as, to prove the strength of gunpowder; to prove the contents of a vessel by comparing it with a standard measure.
 To render certain; put out of doubt (as a proposition) by adducing evidence and argumentation; show; demonstrate.
 To establish the authenticity or validity of; obtain probate of: as, to prove a will. See probate.
 To have personal experience of; experience; enjoy or suffer.
 In arithmetic, to ascertain or demonstrate the correctness of (an operation or result) by a calculation in the nature of a check: as, to prove a sum.
 In printing, to take a proof of.
 Synonyms To verify, justify, confirm, substantiate, make good, manifest.
 To make trial; essay.
 To be found or ascertained to be by experience or trial; be ascertained or shown by the event or something subsequent; turn out to be: as, the report proves to be true; to prove useful or wholesome; to prove faithful or treacherous.
 Hence To become; be.
 To succeed; turn out well.
 To thrive; be with young: generally said of cattle.
 n. An obsolete form of proof.
 In homeopathic practice, to test the therapeutic action of (a drug) by observing the symptoms following its administration in appreciable amounts to persons in health.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 v. increase in volume
 v. be shown or be found to be
 v. obtain probate of
 v. prove formally; demonstrate by a mathematical, formal proof
 v. cause to puff up with a leaven
 v. provide evidence for
 v. put to the test, as for its quality, or give experimental use to
 v. establish the validity of something, as by an example, explanation or experiment
 v. take a trial impression of
Etymologies
Examples

They write letters to prove that they "don't count," and they _prove it_. '

It is said of kings and rulers, they must prove that they have a heart, and it may also be said of the man who has no religion, that _he must prove_ that he has a _conscience.

The word prove is usually reserved for mathematics: “to verify the correctness or validity of by mathematical demonstration or arithmetical proof” Random House Unabridged Dictionary.

The voyage is long and dreary — let us hope the boat will not again prove leaky — if so — Lithe not Styx — be the River for me.

Beck, Limpballs and Malkin prove just how evil and cold blooded they are.

What im saying isnt to give up on this now, but to freakin prove me wrong.
EXTRALIFE – By Scott Johnson  Really interesting viewing here…

When people use the words “SmootHawley” today, they usually mean them as a warning that any interference with trade, especially by the United States, could again prove disastrous.

All the polls prove is that the masses can easily be fooled.
CNN Poll: President's approval on health care under 50 percent

What you did prove is how low your standards are for trying to present the truth.
The Volokh Conspiracy » Competing Explanations for the Oppressive Nature of Socialism

The only thing Fux ratings prove is that there are a lot of stupid and gullible people in the world.
Think Progress » Palin’s talk show debut garners weak reviews, lowerthanexpected ratings.
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