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In this paper, we reconsider the well-known result of Pego-Weinstein\cite{MR1289328} that soliton solutions to the Korteweg-deVries equation areasymptotically stable in exponentially weighted spaces. In this work, werecreate this result in the setting of modern well-posedness function spaces.We obtain asymptotic stability in the exponentially weighted space via aniteration argument. Our purpose here is to lay the groundwork to use the$I$-method to obtain asymptotic stability below $H^1$, which will be done in asecond, forthcoming paper \cite{PR}. This will be possible because theexponential approach rate obtained here will defeat the polynomial loss intraditional applications of the $I$-method \cite{MR1995945}, \cite{MR1951312},\cite{pigottorb}.

Asymptotic Stability for KdV Solitons in Weighted Spaces via Iteration