"she saw me sleeping." which parts of speech is "sleeping" in this sentence? ?

Janine studies to improve her math skills

Examine the table of contents. table of contents fear and its physical effects mental effects emotional effects just say "no! " to breathing exercises meditation yoga free from living free seeking support staying sane index in which chapter would a reader find information on seeking support? fear and its effects free from fear just say "no! " to fear seeking support

Dogs make wonderful pets because they us stay happy and healthy. the topic sentence is "dogs make wonderful pets." , is this correct? i keep doubting myself.