What are the hadith texts restricted to authentic hadiths, instead of texts that contain weak hadiths such as Al-Arba'in An-Nawawiyyah and Bulugh Al-Maram? And is there a necessity to memorize texts that include weak hadiths?

A weak hadith is not used as a proof for what is permissible and what is forbidden, and scholars differ regarding acting upon it in matters of virtues. Weak hadiths are divided into three categories: those that are definitively rejected, those that are possibly authentic despite their weakness, and those over which there is a difference of opinion between weakness and acceptability (hasan). The reason why some imams included weak hadiths in their books is due to several reasons, including: believing in their authenticity or acceptability (hasan), strengthening them with corroborating narrations and supporting evidence, clarifying their weakness, acting upon weak hadiths in matters of virtues, the possibility of their authenticity, or because their weakness is minor. The mentioned books (Al-Arba'in an-Nawawiyyah, Bulugh al-Maram, Kitab at-Tawhid) are great and beneficial, and the weak hadiths within them are few and do not diminish their stature. Memorizing them is beneficial for the student of knowledge. The imams benefited from their knowledge of the conditions of narrators and their memorization of authentic and weak chains of transmission to clarify their status. It is recommended to memorize Al-Umdah and then Al-Bulugh for those who seek to memorize hadith texts.