  Let $F$ be a local field of characteristic $p>0$. By adapting methods of
Scholze, we give a new proof of the local Langlands correspondence for $\GL_n$
over $F$. More specifically, we construct $\ell$-adic Galois representations
associated with many discrete automorphic representations over global function
fields, which we use to construct a map
$\DeclareMathOperator{\rec}{rec}\pi\mapsto\rec(\pi)$ from isomorphism classes
of irreducible smooth representations of $\GL_n(F)$ to isomorphism classes of
$n$-dimensional semisimple continuous representations of $W_F$. Our map $\rec$
is characterized in terms of a local compatibility condition on traces of a
certain test function $f_{\tau,h}$, and we prove that $\rec$ equals the usual
local Langlands correspondence (after forgetting the monodromy operator).
