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Shannon's source coding theorem states the following:

$n$ i.i.d. random variables $X_1,\dots,X_n$ each with entropy H(x) can be compressed into more than n⋅H(x) bits with negligible risk of information loss, as n→∞; conversely if they are compressed into fewer than n⋅H(x) bits it is virtually certain that information will be lost.

I was thinking about the easy case when the $X_1,\dots,X_n$ are uniformly distributed on the integers $[1,n]$. Now suppose I want to transmit a value of $(X_1,\dots,X_n)$. Clearly, each $X_i$ has entropy $\log n$ and thus, if we were to send $o(n\log n)$ bits, we would lose information about some of the $X_i$, with high probability $1 - o(1)$.

The way to argue the Theorem for this case, is that it $\log n$ bits suffice to reconstruct any $X_i$, and conversely, if you receive, for example, only $(1/4)n\log n$ bits, then, you would need to correctly guess $3/4 n\log n$ bits, which happens with probability of only $2^{-3/4n\log n}$. Is my reasoning correct?

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The theorem you quote is not stated formally, so it is in fact impossible to prove it. That said, the idea behind the source coding theorem is that a variable of entropy $H(X)$ behaves (roughly) as if it was uniformly distributed on $2^{H(X)}$ values. When the distribution of $X$ is uniform, the theorem becomes trivial, once stated formally. So it's a good idea for you to look up a formal statement of the theorem, and then verify that it indeed holds for uniformly distributed random variables.

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