Q:

Convergence in $L^p$, Cauchy in $L^\infty$

If $u_n$ is a convergent sequence in $L^p$ with $u_n \to u$, and $u_n$ is convergent is $L^\infty$, is it true that the limit in $L^\infty$ must be $u$?  Is it true if $u_n$ are all test functions, i.e. compact support and $C^\infty$? 
I am trying to understand this in order to understand an approximation argument in the Sobolev space $W^{1,\ p}(\mathbb{R})$.

A:

So convergence in $L^p$ implies there exists subsequence $\{u_{n_k}\}$ converging pointwise a.e.  to $u$, and convergence in $L^\infty$ implies $\{u_{n_k}\}$ converges uniformly (and hence pointwise) a.e.  to some $v$, so that $v=u$ a.e.