section_path: Properties/Inequalities
section_name: Inequalities
paragraph_text: Inequalities
  
  The number e is the unique real number such that (1 + 1 x) x < e < (1 + 1 x) x + 1 {\displaystyle \left(1+{\frac {1}{x}}\right)^{x}<e<\left(1+{\frac {1}{x}}\right)^{x+1}} for all positive x.
  
  Also, we have the inequality e x ≥ x + 1 {\displaystyle e^{x}\geq x+1} for all real x, with equality if and only if x = 0. Furthermore, e is the unique base of the exponential for which the inequality a ≥ x + 1 holds for all x. This is a limiting case of Bernoulli's inequality.