Derivation — Amortized Analysis of Union-Find: the Inverse Ackermann Bound

The previous lesson stated, without proof, that union by rank plus path compression together give $O(m\,\alpha(n))$ total cost for $m$ operations, where $\alpha$ is the inverse Ackermann function. This derivation sketches why — reusing the potential-method machinery from Topic 2, applied here to its most dramatic payoff in the entire unit. CLRS itself marks this proof "starred" (its most technically demanding, requiring an auxiliary function CLRS calls the "Ackermann-style" hierarchy); this note gives the structure of the argument at the depth appropriate for a unit-level treatment, without reproducing every algebraic step of CLRS's own multi-page derivation.