 Solution

After a record of value $v$, the first subsequent observation exceeding it has the original distribution conditioned on $X>v$. Indeed, summing over the number of failed observations gives the densityIndependence makes this conditional transition depend only on the latest record. Multiplying these transitions, starting with the density $f(v_1)$, provesThe ordering indicator is part of the density; it is zero elsewhere. Continuity of the observation distribution avoids ties.