Fermat's Last Theorem for n >= min(a,b)

The Last Theorem of Fermat resp. Fermat-Wiles says that the equation a^n + b^n = c^n has no solution for natural a,b,c > 0 and n > 2.
Here's a proof for sufficiently large n. Without loss of generality be a>b, n>=b.

The idea is that at sufficiently large exponents n the term a^n + b^n lies between the n-th powers of two successive natural numbers and therefore can't be the n-th power of another natural number c.

Pascal's Triangle for (a+b)^n

0                  1
1                 1 1
2                1 2 1
3               1 3 3 1
4              1 4 6 4 1

(a+b)^4 = 1*a^4*b^0 + 4*a^3*b^1 + 6*a^2*b^2 + 4*a^1*b^3 + 1*b^4

(a+b)^n = a^n + n*a^(n-1)*b + ... + b^n

(a+1)^n = a^n + n*a^(n-1)   + ... + 1

a^n < a^n + b^n < (a+1)^n