= Solution
Choose linearly independent basis functions $\phi_1,\ldots,\phi_N$ in the <zero-trace Sobolev space>, and put $V_N=\operatorname{span}\{\phi_i\}$. A concrete conforming <Galerkin method> uses continuous piecewise-linear hat functions on an interval mesh, with zero endpoint values. Seek $u_N=\sum_j c_j\phi_j$ satisfying
$$
a(u_N,v_N)=\langle f,v_N\rangle\quad(v_N\in V_N).
$$
Testing against each basis function gives
$$
\boxed{\sum_{j=1}^N K_{ij}c_j=F_i,\qquad K_{ij}=\int_0^1[p\phi_j'\phi_i'+q\phi_j\phi_i]dx,\qquad F_i=\int_0^1f\phi_i\,dx.}
$$
The matrix is symmetric, and for a nonzero coefficient vector $c$, its represented function is nonzero by basis independence. The <coercivity> estimate gives $c^TKc=a(u_N,u_N)>0$. Thus $K$ is a <positive-definite matrix>, the finite-dimensional problem has a unique solution, and it minimizes the same energy over $V_N$. This is <Ritz-Galerkin equivalence for a symmetric coercive form>.
Subtracting the continuous and discrete equations proves <Galerkin orthogonality>: $a(u-u_N,v_N)=0$ for all $v_N\in V_N$. Let $\|v\|_a=\sqrt{a(v,v)}$. For any comparison $v_N$, orthogonality and the <Cauchy-Schwarz inequality> in this inner product give
$$
\|u-u_N\|_a^2=a(u-u_N,u-v_N)\leq\|u-u_N\|_a\|u-v_N\|_a.
$$
Therefore
$$
\boxed{\|u-u_N\|_a\leq\inf_{v_N\in V_N}\|u-v_N\|_a.}
$$
With $C_a=p_{\max}+q_{\max}/\pi^2$, the same argument in the derivative norm gives the <Céa lemma> estimate $\|u-u_N\|_H\leq(C_a/p_{\min})\inf_{v_N\in V_N}\|u-v_N\|_H$.
Convergence now follows from approximation, not from a bare assertion that the residual is small. The defining density of smooth compactly supported functions in $H_0^1$ lets one approximate $u$ by such a smooth function. Its piecewise-linear interpolant has zero endpoints and derivative error tending to zero as the largest mesh interval tends to zero. For a smooth $w$, the derivative on each element is the average of $w'$ there, and the interval <Poincaré inequality> bounds its error by the element length times $\|w''\|_2$. Density and the boxed best-approximation estimate consequently give $u_N\to u$ in $H_0^1$, and hence in $L^2$, without assuming a bounded derivative of $p$. If the stronger regularity $u\in H^2$ is available, the same interpolation estimate gives first-order convergence in the derivative norm.
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