Tangibility of the transcendental

I was doing a bit of reading about transcendental numbers today, and had an idea of constructing my own well defined transcendental. I was inspired by the Liouville constant.

Anyway, my idea was to construct a number which is a base 10 concatonation of the digits of consecutive prime numbers:
m_p =  0.2357911131719232931...
or the natural numbers:
m_N = 0.12345678910111213...
The construction proceeded as follows. First we define the function lambda:

Now we can express the numbers as infinite sums:

Where p_n is the n'th prime.

Initially I thought this to be an exercise in creative procrastination, since firstly the idea is dead simple, and secondly who would be silly enough in going to the trouble of formulating such numbers. Surely only a fresh graduate with too much time on his hands.

To my surprise the idea behind those numbers has been already formulated by not one, but a few mathematicians. My m_N is in fact a constant previously formulated by D. G. Champernowne. And it turns out that m_p is already known as the
Copeland-Erdos constant:

Notice the striking similarity in the formulation - I have formulated m_p completly oblivious to the Copeland-Edros formulation. The similarity lies in the fact that my lambda function acts a lot like the floor of base 10 logarithm of y, plus 1. In fact the relation is the following:

Which makes the sums in the exponents of the definitions of m_p and the Copeland-Edros constant equal, since for each i we subtract 1 i.e. we subtract 1, k times, we need to add k back to the sum.

Well, I guess I feel better knowing that I'm in good company deeming quite obscure and prima facie pointless ideas valuable. I knew that both m_N and m_p are irrational, but now I also know that m_N is in fact transcendental (mission accomplished!), and hence don't have to provide a proof, which i feel would be a herculean task.