One can find several definitions of the concept of a Cauchy sequence in a fuzzy metric space in the sense of Kramosil and Michalek. This paper presents an example illustrating that definitions given by M. Grabiec
[ibid. 27, 385-389 (1989; Zbl 0664.54032
)] and R. Vasuki
[ibid. 97, 395-397 (1998; Zbl 0960.54005
)] are not equivalent to the definition of George and Veeramani. This fact explains in particular why “fixed point type” theorems proved by different authors (see e.g., papers by M. Grabiec, V. Gregori and A. Sapena and R. Vasuki) sometimes have a different “shape”.