Proof of        ⌊(Subscript[log, 2])n⌋ + 1 = ⌈(Subscript[log, 2])(n+1)⌉

Function (2^n) is an increasing function

x<y -> (2^x)<(2^y) (* by definition of increasing function *)
x≤ y -> (2^x)≤ (2^y) (* by definition of non-decreasing function *)

Function (Subscript[log, 2]) n is an increasing function

x<y -> (Subscript[log, 2])x<(Subscript[log, 2])y (* by definition of increasing function *)
x≤ y -> (Subscript[log, 2])x≤ (Subscript[log, 2])y 
(* by definition of non-decreasing function *)