Let $f:M \rightarrow \mathbb{ R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{ R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is defined by counting cascades \cite{ FraTheA}. Similar data, which also includes a parameter $\epsilon > 0$ that scales the Morse-Smale functions $f_j$, can be used to define an explicit perturbation of the Morse-Bott function $f$ to a Morse-Smale function $h_\epsilon:M \rightarrow \mathbb{ R}$ \cite{ AusMor} \cite{ BanDyn}. In this paper we show that the Morse-Smale-Witten chain complex of $h_\epsilon$ is the same as the Morse chain complex defined using cascades for any $\epsilon >0$ sufficiently small. That is, the two chain complexes have the same generators, and their boundary operators are the same (up to a choice of sign). Thus, the Morse Homology Theorem implies that the homology of the cascade chain complex of $f:M \rightarrow \mathbb{ R}$ is isomorphic to the singular homology $H_\ast(M;\mathbb{ Z})$.
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