Will, present continuous, to be going to

Question

1. Sally is meeting John at seven o’clock this evening in a restaurant downtown.

2. A: Are you busy this evening? B: Yes. I’m going to meet Jack at the library at seven. We’re going to study together.

Why the writer has used "present continuous" for 1 above and "be going to" for the 2 above while the place and time of the event are given.

3. In Section 2 we will prove that.... (in an article)

Is that a correct sentence? Or we should say "In section 2 we are going to prove that...?

Answer 1

For scheduled or timetabled events, we can use present simple:

My flight leaves at 7:35.

We can also use present continuous for planned events or activities, and when an activity may go on for some time:

Sally is meeting John tonight.

The use of going to is less formal, and suggests that it's a personal intent rather than a timetabled event:

I'm going to be a pilot when I grow up

We use will :

• when we don't know when something happen: One day I will be able to ...
• to formally express an intent: In section 2, we will prove that ...
• to express opinions: It will probably rain tomorrow
• to express willingness: I will help you with the washing up
• to make offers and promises: I will send you that file tomorrow

You can find out more about ways to talk about the future in this British Council article.

Answer 2

If someone were to tell me "I'm going to meet Jack at the library at seven," I would understand this to mean speaker 'B' is meeting with Jack at 7 o'clock.

One might argue that "I'm going to meet Jack..." emphasizes the act of "going". But, fluent speakers probably would not dwell on this emphasis too much. As a result, other similar constructions that would convey a similar meaning include:

"I'm going to be meeting with Jack at 7"

"I'm meeting with Jack at 7"

"I'm meeting Jack at 7"