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According to this answer it is possible for galaxies' light to move beyond the visible frequencies due to redshift:

It is possible that eventually the light from them could move into the infrared and even the microwave in extreme cases

Could this have already happened? Have we already looked for it? Is there a telescope with the instrumentation to detect galaxies' 'light' at these wavelengths?

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    $\begingroup$ If the visible light they emit is shifted into the infrared, then any UV they emit will be shifted into the visible. $\endgroup$ Commented Dec 3, 2014 at 21:41
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    $\begingroup$ @KeithThompson That is true of course, though usually the UV light is heavily attenuated by dust absorption. This is the principle behind finding high redshift galaxies via the "Lyman break" technique. $\endgroup$
    – ProfRob
    Commented Dec 4, 2014 at 7:34

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Yes, of course. Many, many examples. Telescopes work in the infrared, far-infrared and there are even samples of galaxies that are selected on the basis of their mm emission.

The most distant galaxies detected now have redshifts of 10 or more (see for example here). This means the wavelength of their light has been stretched by a factor $1+z$ - i.e. by a factor of 11. Thus light in the visible range, say 500nm, now appears at wavelength of 5.5 microns, in the infrared.

Telescopes that work in this range include the Spitzer space telescope; the James Webb Space Telescope and many ground-based telescopes. Observations of highly redshifted galaxies are routinely made at infrared wavelengths on telescopes all around the world.

Galaxies are also detected in the far infrared by the Herschel satellite or at mm (getting on for microwave) wavelengths by JCMT or the ALMA telecope.

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