You have to call TMemIniFile.UpdateFile
to save the information into the file.
Documentation is very clear to that
TMemIniFile buffers all changes to the INI file. The INI file is read once, when the object is first created. Data from the INI file is stored in nested string lists. Each section in the INI file occupies one element in the top-most string list, and each element in this may itself contain a string list. Each element in each of the contained string list represents a key within the section. After the data is read, any changes to the data are stored in memory. To write the data from memory back to the associated INI file, call the UpdateFile method.