Hyers-Ulam stability of isometries on bounded domains

More than 20 years after Fickett attempted to prove the Hyers-Ulam stability of isometries defined on bounded subsets of $\mathbb{R}^n$ in 1981, V\"{a}is\"{a}l\"{a} improved Fickett's result significantly. In this paper, we will improve Fickett's theorem by proving the Hyers-Ulam stability of isometries defined on bounded subsets of $\mathbb{R}^n$ using a more intuitive method different from that used by V\"{a}is\"{a}l\"{a}.