Do the digits(after comma) of the irrational number $\sqrt{2}$ contain all natural numbers without jumping order?

Question

If we mention about $\sqrt{2}=1.41421356$ It contains 1,2,3,4,5,6,41,414,4142,41421 But it does not contain 4121 $\sqrt(2)$ does not contain repetitive patterns and it is not transcendental number(0,10010001...)(which contains only specific digits) and I tried to prove that after comma after some point $\sqrt(2)$ can not contain repetitive natural number with specific distance such as(3165,3,3165,7,3165,9,3165,3,..)and I tried to prove that after comma at some specific point some digit can not be eliminated such as 3 or 2 are possible but others not(3264,3,2563,2,3564,3,...).if these two sentences are proved.İt can prove that $\sqrt(2)$ contains all natural numbers.In fact I tried to prove that impossibility of solving this problem(I think that in math there can be propositions which can be true or false but proving is impossible.)