Answer 1
It took mankind tens of thousands of years to figure out that air existed, let alone that it was matter. It was only in recent human history that we figured out anything about air. Proving that air is matter is analogous to today's physics experiments where you cannot see the object of your study, but have to define its properties and its existence from indirect evidence.
We define matter as something which occupies space, is effected by gravity and has weight. Make a vessel that won't collapse if there is no air inside of it. Weigh the vessel when it is full of air. Then pump all of the air out and weigh the vessel again. The difference in weight is the weight of the air.
There is a famous experiment done by Otto Von Guericke in 1654 in Regensburg, Germany. Regensburg was a Roman outpost on the banks of the Danube River. If you ever go there, I highly recommend the Wirstkuke, an 850 year old restaurant near the river. It was there when Guericke was studying air and he may have had a dinner or two there. Anyway, to prove that air exists and has pressure he made a hollow sphere made of two copper halves and sealed it with a gasket. He used an air pump, which he also invented, to pump the air out of the sphere. Air pressure held the two halves of the sphere together. He then took two teams of horses and had them try to pull the sphere apart. They failed. Guericke then opened a valve that let the air back in and that is when the sphere fell apart under its own weight. The sphere was 14" in diameter meaning the air pressure exerted a force of approximately 4.5 tons.
The force would have been the same if one side of the sphere was attached to something fixed, like a really big rock, instead of another team of horses. Guericke might not have understood that or he might have just appreciated the drama of using two teams. Showmanship, you see, is important even in science.
Answer 2
Matter is anything that has mass and takes up space. So, in order to prove that air is matter, we need to prove that air has mass and takes up space. It's easier to prove that air takes up space, so let's do that part of the problem first.
Go and get a balloon. While you're at it, get two balloons. Go ahead and inflate the balloons with air. The balloons get larger as you put air into them. The only way that air could make them get larger is if air takes up space, so half of our proof is complete. Tie the balloons closed so that they stay inflated - we will need both balloons for the second half of this problem.
Although air has mass, a small volume of air, such as the air in the balloons, doesn't have too much. Air just isn't very dense. We can show that the air in the balloon has mass by building a balance. For this, you will need a meter stick, some tape, some string and a sharp needle. Take some of the string and tie one end to the middle of the meter stick. Take the other end of the string and tape it to the top of a table or a counter, just make certain that the meter stick is free to move around. Tie a section of string to each balloon. On one balloon, make an "X" with two pieces of tape (if you want to be fair, you can make a tape "X" on the second balloon as well, but we really only need one). Take the balloons and tie each one to the meter stick, one on each end of the meter stick. Balance the meter stick by repositioning the balloons, if necessary.
So, at the moment, you should have two balloons hanging from a meter stick, one from each end. If one of the balloons changes mass, we will be able to tell because the meter stick will 'tilt' towards the more massive object. So, all you need to do is to let the air out of one of the balloons. Take the needle and CAREFULLY poke a hole in the center of the "X". You don't want to pop the balloon - you just want to make a hole so that the air will leak out. Hopefully, the tape will keep the balloon together...
What happened? If all went well, one balloon lost its air in a very calm, controlled fashion without sending its balloon guts all over the room. The end of the meter stick with the deflated balloon should have risen into the air. It did this because there was less mass in the balloon after it deflated. The only way the balloon could have lost mass is if the air that was inside it has mass.
With this experiment you have shown that air takes up space and has mass, so you have proven that air is matter.