In the derivation of Euler-Lagrange equation, as given here (under the "Statement" section, with the name "Derivation of one-dimensional Euler–Lagrange equation"), at the end of the derivation, it is assumed that the arbitrary pertubation path $\eta(t)$ has compact support in order to use the fundamental lemma of calculus of variations.
However, is there any justification for this assumption? Can't a path have non-compact support? If so, why?
The derivation takes place on the compact interval $[a,b]$. Any function whose domain is a compact space automatically has compact support.
Since the support of a function is the closure of the set where the function is not $0$, it is a closed subset of a compact space, and is therefore compact itself. The function doesn't even need to be continuous for this to be true.