  We establish the nonlinear stability of $N$-soliton solutions of the modified
Korteweg-de Vries (mKdV) equation. The $N$-soliton solutions are global
solutions of mKdV behaving at (positive and negative) time infinity as sums of
$1$-solitons with speeds $0<c_1<\cdots< c_N$.The proof relies on the
variational characterization of $N$-solitons. We show that the $N$-solitons
realize the local minimum of the $(N+1)$-th mKdV conserved quantity subject to
fixed constraints on the $N$ first conserved quantities.To this aim, we
construct a functional for which $N$-solitons are critical points, we prove
that the spectral properties of the linearization of this functional around a
$N$-soliton are preserved on the extended timeline, and we analyze the spectrum
at infinity of linearized operators around $1$-solitons. The main new
ingredients in our analysis are a new operator identity based on a generalized
Sylvester law of inertia and recursion operators for the mKdV equation.
