= Solution
Let $A_h$ be the symmetric Dirichlet second-difference matrix. Its <eigenvalues> are nonpositive, so every eigenvalue of
$$
R_h=(I-kA_h/2)^{-1}(I+kA_h/2)
$$
has modulus at most one. Thus $\|R_h\|_{2,h}\leq1$.
Insert the exact smooth solution into the scheme. The <local truncation error> per step has discrete norm $O(k^3+kh^2)$, from the order-three trapezoidal time residual and the order-two centred spatial residual. If $e^n$ is the grid error, stability and iteration give
$$
\|e^{n+1}\|_{2,h}\leq\|e^n\|_{2,h}+C(k^3+kh^2).
$$
There are at most $1/k$ steps and $e^0=0$, hence
$$
\max_n\|e^n\|_{2,h}\leq C(k^2+h^2)\to0.
$$
This proves convergence for every $μ=k/h^2>0$ without invoking the Lax equivalence theorem.
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