What does rate of convergence in probabilty theory?

I know that the sample mean converges to population mean with convergence rate $\frac{1}{\sqrt{n}}$.

How can we prove this?

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How do you know this if it has not been proved to you? Proof by higher authority? – Dilip Sarwate Oct 19 '12 at 12:38
@DilipSarwate I think that your comment is not warranted in the context of someone actually asking for an explanation because they are precisely not content with a statement of fact. I might add that I know that the finite simple groups have been classified, that the axiom of choice is independent of the other ZF axioms, etc. Do you know any of these things? – Phira Nov 3 '12 at 15:24
I don't understand the NARQ votes. Although the question could use a little more meat on its bones, it is quite clear what is being asked. – Emily Nov 3 '12 at 15:42
@Phira Having taught probability theory to undergraduates for close to forty years, let me assure that many students who claim to "know" things about convergence in probability theory (because they have copied it down from the blackboard or read it in their textbooks) would be hard put not just to distinguish between the various forms of convergence but even to state correctly, even after prompting and gentle nudging, the definition of a convergent sequence of real numbers. There is no point answering this question unless the OP reveals something about the level of his knowledge. – Dilip Sarwate Nov 4 '12 at 15:55