Hadith: With the start of the last ten days of Ramadan, the Messenger of Allah (may Allah's peace and blessings be upon him) would pray all night, wake up his family, show diligence (in worship), and tie up his lower garment firmly

عن عائشة رضي الله عنها قالت: كان رسول الله صلى الله عليه وسلم إذا دَخَلَ العَشْرُ أَحْيَا الليلَ، وأَيْقَظَ أَهْلَهُ، وَجَدَّ وَشَدَّ المِئْزَرَ.
[صحيح] - [متفق عليه]

‘Ā’ishah (may Allah be pleased with her) reported: With the start of the last ten days of Ramadan, the Messenger of Allah (may Allah's peace and blessings be upon him) would pray all night, wake up his family, show diligence (in worship), and tie up his lower garment firmly.
Sahih/Authentic. - [Al-Bukhari and Muslim]

Explanation

When the last ten nights of Ramadan would come, the Prophet (may Allah's peace and blessings be upon him) would enliven the whole night by performing different types of acts of obedience. He would wake his family up to pray and exert himself in doing more acts of worship than what he usually did. He would devote himself entirely to worship and avoid having intimate relations with his wives.