Let T be a bounded linear operator on a Banach space X. And let ${{ ho}T}(X)$ be the local resolvent set of T at $x;{in};X$. Then we prove that a complex number ${lambda}$ belongs to ${{ ho}T}(X)$ if and only if there is a sequence ${x_{n}}$ in X such that $x_n;=;(T - {lambda})x_{n+1}$ for n = 0, 1, 2,..., $x_0$ = x and ${{parallel}x_n{parallel}^{frac{1}{n}}}$ is bounded.