  Let $^{(r,s)}X_t$ be the L\'evy process $X_t$ with the $r$ largest positive
jumps and $s$ smallest negative jumps up till time $t$ deleted and let
$^{(r)}\widetilde X_t$ be $X_t$ with the $r$ largest jumps in modulus up till
time $t$ deleted. Let $a_t \in \mathbb{R}$ and $b_t>0$ be non-stochastic
functions in $t$. We show that the tightness of $({}^{(r,s)}X_t - a_t)/b_t$ or
$({}^{(r)}\widetilde X_t - a_t)/b_t$ at $0$ implies the tightness of all normed
ordered jumps, hence the tightness of the untrimmed process $(X_t -a_t)/b_t$ at
$0$. We use this to deduce that the trimmed process $({}^{(r,s)}X_t - a_t)/b_t$
or $({}^{(r)}\widetilde X_t - a_t)/b_t$ converges to $N(0,1)$ or to a
degenerate distribution if and only if $(X_t-a_t)/b_t $ converges to $N(0,1)$
or to the same degenerate distribution, as $t \downarrow 0$.
