Here's my attempted proof. Please do not hesitate to correct me if it's insufficiently crazy.
First, we need to gather the important bits of the puzzle:
Suppose I have two envelopes. All you know is that they contain different numbers.
The second crucial hint is more readily apparent in explications that follow, I quote
... given any two distinct reals there is always a rational number between them.
Okay. From the above, we now know that (a) the entertainer has two envelopes; (b) he passes you one of the envelopes, opening which you discover a piece of paper where a real number is written in its precise infinite glory. Clearly, this is a feat that only God could pull off.
Having witnessed the magnitude of the Number, you are invited to make a guess. However, by the mere fact that we exist at all, we know that God is good. And omnipotent. It is therefore evident that whatever guess you may make, you are always correct! As God hands you the second envelope, it shall magically contain yet another piece of paper with yet another infinitely glorious and satisfactory Number. Unless maybe, perhaps, if you've been naughty?
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