What you suggest is logical, as is the Hungarian sentence. But the expression in English is, for whatever reason, 'in' rather than 'into'.
Fixed now. 'Writing into this book" is not common usage. But does happen typically as part of a phrase. Perhaps you are exasperated with someone asking you to make sure you remember something and you reply, "I will remember it! Look! I am writing into this book, 'Must learn Hungarian every day!'. See? I've written it just there!"
(I don't recommend this stroppy tone of voice with your friends and loved ones ;-)
Does this imply that the book is a journal, or that maybe the speaker is writing notes in the margins of a book? That is, is the speaker writing on the pages of the book?