  Extreme mass ratio inspirals (EMRIs) take place when a stellar-mass black
hole (BH) merges with a supermassive black hole (SMBH). The gravitational wave
emission from such an event is expected to be detectable by the future Laser
Interferometer Space Antenna (LISA) and other mHz detectors. It was recently
suggested that the EMRI rate in SMBH binary systems is orders of magnitude
higher than the EMRI rate around a single SMBH with the same total mass. Here
we show that this high rate can produce thousands of SMBH-BH sources at a
redshift of unity. We predict that LISA may detect a few hundred of these EMRIs
with signal-to-noise ratio above SNR>=8 within a four-year mission lifetime.
The remaining sub-threshold sources will contribute to a large confusion noise,
which is approximately an order of magnitude above LISA's sensitivity level.
Finally, we suggest that the individually detectable systems, as well as the
background noise from the sub-threshold EMRIs, can be used to constrain the
SMBH binary fraction in the low-redshift Universe.

High Energy Astrophysical Phenomena;Cosmology and Nongalactic Astrophysics;Astrophysics of Galaxies