  The sensitivity of future electron-proton colliders, the LHeC and FCC-eh, to
weakly-produced supersymmetric particles is evaluated in this article.
Supersymmetric scenarios where charginos ($\tilde{\chi}_1^{\pm}$) and
neutralinos ($\tilde{\chi}_1^{0}$ and $\tilde{\chi}_2^{0}$) are nearly
degenerate in mass are considered. Two sets of models, which differ in the mass
of sleptons ($\tilde{\ell}$), are studied. Under the hypothesis that slepton
masses are at the multi-TeV scale (``decoupled'' scenario), the production
processes for charginos and neutralinos at $ep$ colliders, $p\, e^- \to j\,
e^-\, \tilde{\chi} \tilde{\chi}$ with $\tilde{\chi}=\tilde{\chi}_1^{0}$,
$\tilde{\chi}_1^{\pm}$ or $\tilde{\chi}_2^{0}$, are considered. For the models
where slepton masses are above but close to $\tilde{\chi}_1^{\pm}$,
$\tilde{\chi}_2^{0}$ masses (``compressed'' scenario), contributions from the
processes $p\, e^- \to j\, \tilde{\chi}\, \tilde{e}_L^-$ and $j\,
\tilde{\chi}\, \tilde{\nu}$ followed by the decays $\tilde{e}_L^- \to
\tilde{\chi}_{1,2}^{0} + e^-$ and $\tilde{\nu} \to \tilde{\chi}_1^+ + e^-$ are
also taken into account. These scenarios are analysed with realistic detector
performance, using multivariate techniques. Effects of systematic uncertainties
and electron beam polarization dependence are also discussed. The reach is
found to be complementary to the one obtained at $pp$ colliders, in particular
for the compressed-slepton scenario.
