Firstly you're only getting a little bit closer to the Sun. The Sun is 93 million miles away. This isn't much when you compare it to the 93 million miles to the Sun. To make a real difference, you'd have to get much closer to the Sun - like ten thousand miles closer. It doesn't work quite that simply though.
Hot air rises.
You can prove this by filling a balloon with hot air - the air from your own lungs is warm enough. The balloon will float as long as the air inside it is hotter than the air outside it. Well as the air rises, it cools down. Things need energy to move and the extra energy that the air has in its heat is used up moving it upwards though the atmosphere. Eventually the air will end up at the same temperature as the air around it and will not rise upwards any more.