Clarification on bounded variation of the terms in the Ito's formula as per Ikeda and Watanabe's book

Let $X(t) = X(0) + M(t) + A(t)$ be a continuous semi-martingale where $M \in \mathscr {M}$ and $A \in \mathscr A.$ Let $F: \mathbb R \to \mathbb R$ be $C^2$ be a function of class $C^2$. Then $$F(X(t))=F(X(0)) + \int_{0}^{t} F'(X(s))dM(s) + \int_{0}^{t} F'(X(s)) dA(s) + \frac 12 \int_{0}^{t} F''(X(s))d\langle M \rangle(s) \; \text{a.s}$$ for $t \ge 0.$ Here I want to show that the terms $\int_{0}^{t} F'(X(s)) dA(s)$ and $\int_{0}^{t} F''(X(s))d\langle M \rangle(s)$ belong to $\mathscr A$, i.e, almost all sample functions of them are of bounded variation on each finite interval.

My attempt:

Fix $w \in \Omega$ and $t \ge 0$. Consider $\int_{0}^{t} F''(X(s))d\langle M \rangle(s)$ first. Then $$\sum_{i=1}^{n} |\int_{0}^{t_i} F''(X(s))d\langle M \rangle - \int_{0}^{t_{i-1}} F''(X(s))d\langle M \rangle| \le \sum_{i=1}^{n} |\int_{t_{i-1}}^{t_i} F''(X(s))d\langle M \rangle|.$$ Now $F''$ is continuous on $[0,t]$ and hence bounded. Say the bound is $C_{\omega}$. $$\sum_{i=1}^{n} |\int_{t_{i-1}}^{t_i} F''(X(s))d\langle M \rangle| \le C_{\omega} \sum_{i=1}^{n}|\langle M \rangle_{t_i} - \langle M \rangle_{t_{i-1}}|.$$

Here I got stuck and I am not sure what to do from here.

In the case of the other term $\int_{0}^{t} F'(X(s)) dA(s)$, I follow the similar steps as done above to get that

$$\sum_{i=1}^{n} |\int_{0}^{t_i} F''(X(s))dA(s) - \int_{0}^{t_{i-1}} F''(X(s))dA(s)| \le C_{\omega} \sum_{i=1}^{n}|A_{t_i} - A_{t_{i-1}}|.$$

As $A \in \mathscr A,$ the sum $\sum_{i=1}^{n}|A_{t_i} - A_{t_{i-1}}| \lt \infty.$ Here we have shown that for a fixed $\omega$, $\int_{0}^{t} F'(X(s)) dA(s)$ is of bounded variation after taking supremum over all the partition of $[0,t]$ in the above inequality. But from here how does it follow that $\int_{0}^{t} F'(X(s)) dA(s)$ is of bounded variation $\textbf{a.s}?$ How to get rid of $C_{\omega}$?

Ikeda, Nobuyuki; Watanabe, Shinzo, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, Vol. 24. Amsterdam - Oxford -New York: North-Holland Publishing Company. Tokyo: Kodansha Ltd. XIV, 464 p. $ 85.25; Dfl. 175.00 (1981). ZBL0495.60005.*

Recall that a function is of bounded variation if and only if it is the difference of two non-decreasing functions. For each $\omega \in \Omega$ we have

$$\int_0^t F^{\prime \prime}(X_s) d \langle M \rangle_s = \int_0^t F^{\prime \prime}(X_s)^+ d \langle M \rangle_s - \int_0^t F^{\prime \prime}(X_s)^- d \langle M \rangle_s$$

Thus it follows that the integral on the left is the difference of two increasing functions, as the integrator $d\langle M \rangle_s$ induces a positive measure on $\mathbb{R}$ for each $\omega$.

For the integral involving $A$, argue similarly, but perform a Jordan decomposition for $A_s(\omega)$ beforehand.