  We measured the central ($1/2\leftrightarrow -1/2$) and first satellite
($\pm3/2\leftrightarrow \pm1/2$) lines of the \la\ NMR spectra as a function of
temperature in LaAgSb$_2$, in order to elucidate the origin and nature of the
charge-density-wave (CDW) transitions at $T_\text{CDW1}=207$ K and
$T_\text{CDW2}=186$ K. In the normal state, the Knight shift K reveals a fairly
linear relationship with decreasing temperature, which is ascribed to a
pseudogap in the spin excitation spectrum, pointing towards the material being
an unconventional metal. Upon further cooling, K decreases more steeply below
$T_\text{CDW1}$, indicative of the partial Fermi surface gap opening on top of
the pseudogap. The most remarkable finding in our study is a clear splitting of
the satellite lines at $T_\text{CDW1}$ observed for $H\parallel c$, whose
temperature dependence behaves as the BCS order parameter in the weak-coupling
limit, evidencing that the CDW transition induces the periodic lattice
distortion. Our NMR findings therefore demonstrate that the CDW transition in
LaAgSb$_2$ is of Peierls type, being driven by the electronic instability in
the vicinity of the Fermi level.
