Sayyiduna al-Hajjaj bin Amr al-Mazini (may Allah be pleased with him) says: One of you thinks that whoever prays continuously until morning has offered tahajjud. Rather, tahajjud is that one prays after sleeping, and then prays again after sleeping. This was the manner of the Prophet's (peace and blessings be upon him) prayer.
Al-Tabarani recorded part of it in al-Mu'jam al-Awsat and al-Mu'jam al-Kabir. Some versions contain these words: The Prophet (peace and blessings be upon him) used to offer tahajjud after sleeping (and waking up), and he used to use the tooth-stick before offering tahajjud. This narration revolves around Abdullah bin Salih, the scribe of al-Layth. Abd al-Malik bin Shu'ayb bin al-Layth said about him: He is thiqah (trustworthy) and reliable; Ahmad and others declared him da'if (weak).