It’s very difficult to prove anything, but easier to disprove something. We can disprove Flat Earth without getting off the sofa. This is how :

We can assume three observations of the moon. The reason that we can assume them is that they have been made for hundreds, if not thousands of years by billions of people and are not at all controversial or questioned by anyone of sound mind. Moreover, these observations are easily repeatable by ordinary people without reference to any scientific authority and they are free from tampering, fraud, or any other nefarious activity of real or imagined conspirators.

The three observations are :

1. We see the same moon face from all locations on Earth
2. The moon appears round
3. The orientation of the moon appears to be rotated depending on the observers’ latitude.

1. The Same Face 

The fact that we all see the same face of the moon speaks to its distance from the Earth. Imagine – Two observers (Alice and Bob) on a desert highway getting out of a car to view a distant mountain range in the south across the desert. They walk forward of the car for ten yards. Both observers, being separated by only the width of a car will see the same view of the mountains but when they turn around, their views of the car will be very different; they will see different aspects of the car. The difference in their views is dependent upon their distance from the object and the distance from each other, where it creates a different angle of view of the object. Imagine now that Alice walks east for three miles and Bob walks west for the distance. This six-mile difference should give each observer a unique view of the range.

Observers of the moon in Melbourne in Australia, and Punta Arenas in Chile are a whole world away on the flat earth map. On the globe, they are 9300 miles but on the flat earth much farther apart. If the moon was anywhere near the distance claimed by flat earthers, the views of the moon from these two places would be quite different. They are in fact of the same face.

2. Appearing Round

The fact that the moon appears round means that it’s one of a few shapes. It could be a globe, a disc, a cone, a hemisphere, or a cylinder. It could be an irregular shape with a perfect round face but I don’t think anyone expects that. The fact that when it casts a shadow on the earth, that shadow is always a circle, that it always appears as a circle, and that it has a gradient of light and dark when not full are strong suggestions that it’s a sphere. The fact that all large celestial bodies are spheres and that we know why they tend to be spherical also indicates a spherical moon. But that’s not the point, we are talking about the shape of the earth, this is only important to debunk one of the contrived and ad-hoc explanations expected down the line.

3. Orientation

The difference in the apparent orientation is well explained by a global earth, simply put – From the moon’s point of view, its observers near the Earth’s North Pole appear upside down to its observers near the Earth’s South Pole.  And so those observers’ views of the moon are correspondingly inverted.

4. Conclusions

1. There is no explanation that accounts for the orientation change on flat earth while maintaining the circular appearance of the moon.
2. There is no mathematics needed to realise that the moon must be a distant object because observers thousands of miles apart see the same aspect of the moon whatever its shape may be.
3. The observations are unquestionable, easy to repeat by ordinary people at no cost, and the conclusions are obvious without reference to authority use of mathematics or difficult-to-understand science. Common Sense leads us to reject the flat earth and seek an alternative model. The Globe Model explains all observations perfectly. Other shapes can be discussed but Flat is dead in the water.