Llewellyn's Number

I got the idea for "Llewellyn's Number" upon reading Carl Sagan's only science-fiction book, Contact. In this story, an advanced alien species has discovered that hidden in certain important numbers like pi and e, are clearly messages when properly decoded. Because mathematics cannot be technologically faked, the aliens take this as proof of God's existence. While I'm not interested in such "proofs" myself, the idea of transcendental numbers containing messages intrigued me. I wondered about a more general idea: in a number like pi, would ANY finite-length string of digits occur? If so, then ALL messages, both true and false, would occur using any decoding scheme. It's obvious that this could only be true for finite-length strings. It's also clear that it is not true for all transcendental numbers. For example, the Liouville numbers consist only of 1s with intervening numbers of zeros which strictly increase according to the factorial function. It's also obvious that if pi (let's just take pi as our "classic example") does possess this property, then each finite-length digit string must occur an infinite number of times. This is because if it is a string of length N, it also is the prefix for 10 strings of length N+1, 100 strings of length N+2, etc, and all these longer strings must also occur by supposition. I'm assuming decimal expansion here although any base would do.

Not being able to determine if pi or e has such properties, I decided to construct my own number which surely has this property, so I call it Llewellyn's Number, or L-10 (10 for base 10, there is one such number for every integer >= 2, so there are actually an infinite number of them). L-10 begins with a decimal point, followed by the ten digits, i.e., .0123456789 ... it is then followed by all the two-digit numbers, so the next two hundred digits are 00 01 02... 10 11 ... 98 99 (spaces added for clarity only). So now we have L-10 = .0123456789000102...99 .... Now we follow with all 1000 three-digit numbers, etc. Using any typical encoding scheme (for example, each two digits representing its ASCII value) to convert into readable character data, it's clear that L-10 contains every finite length string of decimal digits, each an infinite number of times; thus it contains every possible decoded text, such as English. So it contains an infinite number of copies of Hamlet, including every possible misspelling, alteration, and so on. Of course, most randomly-chosen substrings will be gibberish, but the idea is still cool.

Now I suspect L is transcendental, but I don't know enough math to prove it! Any help out there?