b is a base of log , b > 1 .

Given inequality is logb(n+c) ≦ logb(n) + d and c ≧ 0 .

We want to prove there is a non-negative value d such that given inequality is satisfied for n ≧ 1 .

logb(n+c) = logb[n(1 + c/n)] = logb(n) + logb(1 + c/n) .

Therefore given inequality becomes

logb(n) + logb(1 + c/n) ≦ logb(n) + d

logb(1 + c/n) ≦ d

So if d satisfies logb(1 + c/n) ≦ d then given inequality is satisfied .

logb(x) is an increasing function and 1 + c/n ≦ 1 + c for n ≧ 1 , so logb(1 + c/n) ≦ logb(1 + c) .

Therefore , if d satisfies log(1 + c) ≦ d then given inequality is satisfied for n ≧ 1 .

Such d is always exist .