Let $X$ be a positive continuous stochastic process. Define the sequence of stopping times $$\tau_n := \inf \{ t:X_t>n \}, \quad n \in \mathbb N$$ and define the sequence of stopped processes $\{ X^{\tau_n} \}_{n \in \mathbb N}$ where $ X^{\tau_n}:=\{X_t^{\tau_n};t \geq 0 \}.$ Let $C>0$ be a fixed constant. I know that $$E[X_t^{\tau_n}]<C \ \quad \text{for all $t$ and $n$} \tag*{(1)}$$ and I have to show that $$E[X_t]<C \ \quad \text{for all $t$} \tag*{(2)}.$$ Does (1) imply (2)? If yes, is the following enough justification?
Since X is a continuous process, $n \rightarrow \infty$ implies $\tau_n \rightarrow \infty.$ Thus, for almost all $\omega,$ $$X_t^{\tau_n}(\omega) = X_{t \wedge \tau_n}(\omega)=X_t(\omega) \quad \text{as $n \rightarrow \infty$}. $$ Therefore, by Fatou's lemma, for each $t$ $$E[X_t] \leq \lim_{n \rightarrow \infty} E[X_t^{\tau_n}]< C.$$