Proving this presupposes that limt -> ∞ U(t) exists. Since it exits, call it Q
Then consider the left hand side
lims -> 0 s ∫ U(t) exp(- st)
Interchanging the operation of taking the limit and performing the integration gives
s ∫ Q exp(-st) dt = s Q ∫ exp(-s t) dt = s Q (1/s) exp(-s t) |0 ∞ = s Q /s = Q
Since Q = limt -> ∞ U(t) This proves what was desired.