User blog comment:Sbiis Saibian/Googology101 - Part I/@comment-1605058-20141023055529/@comment-5982810-20141023151405

"Suppose the conjecture holds for sufficiently large numbers, for all integers above some limit N. Now, let n be the least number which is more difficult to write than every number which is at most N. Then, obviously, n>N, but no number smaller than n is more difficult to define, by choice of n. So we get contradiction and conjecture is wrong."

--LP9

I believe your proof is sound. But I already realized this flaw, hence the additional condition that the number must be "namable" in the way I previously described.