Collatz Sequence Proof Easy Way

Abstract

The document proposes an attempted proof of the Collatz Conjecture by asserting that every Collatz sequence ( S(n) ) eventually reaches the loop ({4,2,1}). The author introduces “Taha’s Collatz Fact 1 (TCF1),” which claims that for any starting value ( n \in \mathbb{N}^+ ), the iterative sequence ( S(n) ) shares the same invariant loop ( IS(n) = {4,2,1} ). The text illustrates this with explicit sequences for ( n = 1 ) through ( n = 7 ), each ending in the standard Collatz cycle. The argument then attempts an induction-based extension: assuming ( IS(r) = {4,2,1} ) for all values up to ( r ), the author analyzes the cases ( r+2 ) even and ( r+2 ) odd, concluding that both lead to the same invariant loop. The document ends with the claim that the induction covers all natural numbers, implying ( IS(n) = {4,2,1} ) for all ( n \in \mathbb{N}^+ ). Cited lines: “S(n) = {a, b, c, … , t} = IS(n)” and “IS(n) = {4,2,1} ∀n ∈ (N_even ∪ N_odd) = N+”.