In the textbook Introduction to Financial Derivatives, the quadratic variation is defined as $$ [g,g](t) := \lim_{\Delta t \to 0^+} \sum_0^t [g(t_{i+1}) - g(t_i)] $$
and the author has just claimed that the quadratic variation of the (standard) Weiner process is linear. $$ [W,W]_t = t $$
Next, they suppose that if $\{X_t\}$ satisfies $d X_t = Y_t dt + Z_t dW_t$, where both $\{Y_t\}$ and $\{Z_t\}$ are adapted to $\{W_t\}$, then the quadratic variation of $\{X_t\}$ is given by $$ d[X,X]_t = Z_t^2 dt $$
Is there a proof for the last claim? My intuition is that the process generated by $Y_t dt$ is of bounded 1st variation.