a) Prove that for x ≥ 1 {\displaystyle {}x\geq 1} the estimate
holds.
b) Prove that the function H ( x ) {\displaystyle {}H(x)} defined by
for x ≥ 1 {\displaystyle {}x\geq 1} is increasing.
c) Prove that 10 ! ≥ e 11 + 1 {\displaystyle {}10!\geq e^{11}+1} .
d) Prove that for the factorial function for x ≥ 10 {\displaystyle {}x\geq 10} the estimate