Let’s take a look at the ませんか construction. For added emphasis you could add the phrase “with me”, but it’s already implied that you are extending an invitation to someone to do something with you, so you don’t absolutely need to say “with me”.
～(verb ません form + か？ ＝ Why don’t you ～？/ Let’s ～！ / Won’t you ～？/ Wouldn’t you ～ ？ / You can use ませんか form to extend an invitation.
Verb present negative form + か？ ⤴
Rising intonation at the end of a sentence indicates a question in Japanese.
Examples：
① Won’t you having lunch with me?
昼ご飯（ひるごはん）を 一緒に（いっしょに） 食べませんか？
② Would you like to go to see the movie this weekend?
週末に（しゅうまつに）映画を（えいがを） 見に行きませんか？
③ Would you like to have some cakes after dinner?
ごはんのあとに ケーキを 食べませんか？
④ Why don’t you join the party next Saturday?
来週の（らいしゅうの）土曜日の パーティに 参加（さんか）しませんか？
⑤ Shall we dance?
踊りませんか（おどりませんか？）
Yukiko
