= Solution
Define the <quadratic covariation> of two <continuous semimartingales> by polarization:
$$
[X,Y]_t=\frac14\bigl([X+Y]_t-[X-Y]_t\bigr).
$$
The available <quadratic variation> convergence theorem and the <polarization identity> imply that the sums of products of simultaneous grid increments converge in <uniform convergence on compacts in probability> to $[X,Y]$. The <quadratic covariation> is a continuous finite-variation process, vanishing at time zero.
For a grid ending at $t_n=h_n\lfloor t/h_n\rfloor$, the elementary algebraic identity
$$
X_{(k+1)h_n}Y_{(k+1)h_n}-X_{kh_n}Y_{kh_n}
=X_{kh_n}\Delta_kY+Y_{kh_n}\Delta_kX+\Delta_kX\Delta_kY
$$
telescopes. The first two sums converge by part (a), with the continuous processes $X,Y$ as integrands; continuous paths are left-continuous and locally bounded. The last sum converges to the <quadratic covariation> by the preceding polarization argument. Finally, $X_{t_n}Y_{t_n}\to X_tY_t$ locally uniformly by continuity. Hence
$$
\boxed{X_tY_t=X_0Y_0+\int_0^tX_s\,dY_s+\int_0^tY_s\,dX_s+[X,Y]_t.}
$$
This is the <Itô product rule>, or stochastic <integration by parts>. Uniqueness of the limit in <uniform convergence on compacts in probability>, followed by continuity, makes the identity hold simultaneously for all $t$ outside one null set. The proof used finite-grid algebra, part (a), and the permitted <quadratic variation> theorem; it did not use the <Itô formula>.
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