tanx＝x²在x＞0时每个区间解的存在性，x＞pi/2时的唯一性与解序列随x增大而单调收敛于x＝（n＋1/2）pi上：一个全新原创证明 (A Novel and Original Elementary Proof on tan(x)=x²: Existence of Solutions in Each Interval for x>0, Uniqueness Beyond π/2, and Monotonic Convergence to (n+1/2)π in Chinese)

This manuscript presents a rigorous, original and entirely elementary proof of the existence, uniqueness, and asymptotic behavior of solutions to the transcendental equation tan(x)=x² for x>0. Unlike classical approaches relying on Taylor expansions, numerical methods, or asymptotic analysis, this work exploits only basic calculus, inequalities, and monotonicity arguments—demonstrating that deep results can be extracted with minimal machinery. Key innovations include:

1. Elementary existence/uniqueness proofs via derivative control and sign analysis (no approximations).

2. Explicit monotonic convergence of solutions to (n+1/2)π, derived without infinite series.

3. A self-contained, inequality-driven framework that bypasses transcendental asymptotics.