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Existing Multi-fidelity Bayesian Optimization (MFBO) methods ignore the convergence behavior of the multi-fidelity surrogate as the fidelity increases, leading to inefficient exploration and suboptimal performance. We introduce CAMO (Convergence-Aware Multi-fidelity Optimization), a principled framework based on Linear Fidelity Differential Equations (LFiDEs) that explicitly encodes convergence of fidelity-indexed outputs and employs a closed-form nonstationary kernel. We rigorously prove the existence and pointwise/uniform convergence to the high fidelity surrogate under mild restrictions and provide new convergence results for general FiDEs using smooth, non-smooth and even non-convex Lyapunov functions, establishing a bridge between MFBO and the theory of subgradient flows in non-smooth optimisation theory. Combined with a fidelity-aware acquisition function, CAMO outperforms state-of-the-art MFBO methods on a majority of synthetic and real-world benchmarks, with up to a four-fold improvement in optimisation performance and a dramatic speed-up in convergence. CAMO offers a tractable and theoretically grounded approach to convergence-aware MFBO.