Let $T$ be some random variable on $\mathbb N$ and define $T_n=\min(T, n)$. Is $T$ a stopping time for the natural filtration of $(T_n)_n$?
I'm pretty sure it's not, since $\{T=n\}=\{T_{n+1} = n\}$, so $T$ is a stopping time for $(T_{n+1})_n$ but not $(T_n)_n$. I'm just worried I might confused about the definition of a stopping time and would like to double check.
Intuitively, you must be able to determine whether or not $\{T<=t\}$ occurred on the basis of knowing $\mathcal{F}_t$, the filtration at time $t$. In your case you need to determine if $\{T\leq n\}$ given the filtration $(T_n)_n$. So suppose you know $T_1,T_2,...,T_n$. You now want to find out if $T\leq n$ has occured. One possibility is $T_k=n$ for $k<n$. and $T_{k+1}=n$. However, another possibility is $T_{n-1}=n-1$, $T_n=n$ and $T_{n+1}=n$. This implies that you cannot accurately determine if $T\leq n$, so $T$ is not a stopping time for $(T_n)_n$.