Local martingale is locally uniformly integrable martingale?

Question

Is a local martingale locally uniformly integrable martingale ?

Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale.

But how can we find a localizing sequence such that the stopped process is a uniformly integrable martingale ?

The solution I gave is $\min (\tau_n , n)$, could somebody please confirm ?

Thanks in advance !