Laudator Temporis Acti

Both Homer and Hesiod call Sleep and Death (Hypnos and Thanatos) brothers:

• Homer, Iliad 16.671-672 (tr. A.T. Murray): Give him [Sarpedon] to swift conveyors to bear with them, / even to the twin brethren, Sleep and Death.
  
• Hesiod, Theogony 756 (tr. Hugh G. Evelyn-White): The other [Night] holds in her arms Sleep the brother of Death.
  

The passage from Homer's Iliad is portrayed on a famous red-figure calyx-krater of around 515 BC, signed by Euxitheos (potter) and Euphronios (painter), now in the Metropolitan Museum of Art in New York City. Likewise the passage from Hesiod's Theogony is the inspiration for a carving on a cedar chest described by Pausanias 5.18.1 (tr. W.H.S. Jones) but no longer extant:

There is a figure of a woman holding on her right arm a white child asleep, and on her left she has a black child like one who is asleep. Each has his feet turned different ways. The inscriptions declare, as one could infer without inscriptions, that the figures are Death and Sleep, with Night the nurse of both.

It is easy to see how sleep and death were associated. A sleeper's breathing is often so shallow as to be hard to detect, and the sleeping body is motionless for long periods of time, like a corpse. On the one hand, poets liken death to sleep, e.g. Hesiod, Works and Days 116 (tr. Hugh G. Evelyn-White), speaking of mortals of the Golden Age:

When they died, it was as though they were overcome with sleep.

On the other hand, they liken sleep to death, e.g. Homer, Odyssey 13.79-81 (tr. Richmond Lattimore):

Upon the eyes of Odysseus there fell a sleep, gentle, / the sweetest kind of sleep with no awakening, most like / death.

We find this idea among the ancient Romans as well, whether by borrowing from the Greeks, common Indo-European origin, or independent development, e.g.

• Cicero, De Senectute 22.80: Now indeed you see that there is nothing so like death than sleep (iam vero videtis nihil esse morti tam simile quam somnum).
  
• Vergil, Aeneid 6.278: Death's blood relative, Sleep (consanguineus leti sopor).
  
• Ovid, Amores 2.9.41: What is sleep but the likeness of cold death? (quid est somnus, gelidae nisi mortis imago?)
  

From this idea it is a small step to the notion of death as the sleep from which there is no awakening:

• Aeschylus, Agamemnon 1451: endless sleep (ateleuton hypnon).
  
• Horace, Odes 1.24.5: perpetual sleep (perpetuus sopor).
  

Even now believers in life after death pray for their dear departed in these words: "Eternal rest grant unto them, Lord." (requiem aeternam dona eis, Domine.)