Original sentence: The implications i) ⇒ ii)  ⇒ iii)  ⇒ iv) are trivial.To prove iv) ⇒ i), we first recall that the holomorphy of a function at a is equivalent to it being analytic at a, i.i)  f is holomorphically extendable over a.ii)  f is continuously extendable over a.iii) There exists a neighborhood of a on which f is bounded.iv)  limz → a(z − a) f(z) = 0
Edited sentence: , we first recall that the holomorphy of a function at  is equivalent to it being analytic at , i.The implications 1.There exists a neighborhood of  on which  is bounded.To prove 4.are trivial.is continuously extendable over.is holomorphically extendable over