Localisation in the proof of Ito's formula I am reading Karatza's and Schreve's book "Stochastic Calculus and Brownian Motion" and I don't understand a strange thing as follows:
Let $X=X_0 + A +M $ be a semimartingale, where $A$ is a continuous process of bounded variation and $M$ is a continuous local martingale. In the proof of Itó's formula, he defines a sequence of stopping times $(T_n)$ by
$$ T_n = \begin{array}{cc}
  \Bigg\{ & 
    \begin{array}{cc}
      0, & |X_0| > n \\
      \inf\{ t \geq 0: |A_t| > n \text{ or } |M_t|>n \text{ or } \langle M \rangle_t >n\}, & |X_0| \leq n.
          \end{array}\end{array}
$$
Then he claims that by considering the stopped process $X^{T_n}$, we can assume that $X_0$ is bounded. I don't understand this, as $X^{T_n}_0 = X_0$ is not necessarily bounded.

The thing is that the formula is trivially true at time $t=0$ so you only have to prove it for the case where $|X_0|<n$, which is why Karatzas and Shreve only consider this case making $|X_0|<n$ a property that $X$ respects.
Best regards 
