I'm trying to wrap my head around why Morse-Smale diffeomorphisms are structurally stable, using the real line as a toy example. Say $f$ is a Morse-Smale diffeomorphism. Proofs for more generalized versions use the fact that a map that is $C^1$-$\epsilon$ close to $f$ will have a unique hyperbolic fixed point for each fixed point of $f$. I'm having trouble understanding why this is the case, and I'm not sure how to prove it. I'm thinking of proving this by contradiction, like assume there exists a fixed point $p$ for a function $g$ that is $C^1$-$\epsilon$ close to $f$ that is not hyperbolic, but I'm stuck after writing the conditions $|f(p) - g(p)| = |f(p) - p| < \epsilon$ and $|f'(p) - g'(p)| = |f'(p) - 1| < \epsilon$.
Any ideas how to proceed?