Step 1: Consider energy storage in a short stub.
A short-circuited transmission line stub of length $l$ less than $\lambda/4$ carries current but has zero voltage at the shorted end. For a short stub, the magnetic energy storage (current-driven) dominates over electric energy storage.
Step 2: Link energy storage to impedance type.
An element that stores predominantly magnetic field energy behaves as an inductor. Its impedance is purely imaginary with a positive sign: $Z = +jX$ (inductive reactance).
Step 3: Verify using the input impedance formula.
$Z_{in} = jZ_0 \tan(\beta l)$. For $0 < \beta l < \pi/2$, $\tan(\beta l) > 0$, so $Z_{in} = +jX$ (purely inductive). \[ \boxed{\text{Purely inductive}} \]