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As a newcomer to cryptography, I'm working on Exercise 2.12 in the book, Introduction to Modern Cryptography.It looks like this:

Using the proof of the theorem that says if $E$ is a perfectly secret encryption scheme, then $\lvert K\rvert \geq \lvert M\rvert$), I've shown that the lower bound for the size of the keyspace is as below: $$\lvert K\rvert \geq (1-\epsilon)\lvert M\rvert$$

But here's the problem: I proved this under the condition that $$\lvert\Pr[M=m\mid C=c]-\Pr[M=m]\rvert\leq\epsilon$$

So I think I should show that this definition is equivalent to the condition given in the exercise.

My question is whether the two definitions are equivalent, and if so, how can I prove it?

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