If $u_n\to u$ in $L^p$ and $u_n\to v$ in $L^q$ do we have that $u=v$?

Question

If $u_n\to u$ in $L^p$ and $u_n\to v$ in $L^q$ do we have that $u=v$ ?

Attempt

I know that we have a subsequence $u_{n_k}$ that converge to $u$ and $v$ a.e. and thus that $u=v$ a.e., but do we have that $u=v$ everywhere ?