The article below is a great example of what you can do if you are trying to exercise, but can't get outside to do it. After a life-long struggle of being overweight, the woman in the article below was determined to make changes in efforts lose weight. The first change was diet; this would prove to pave the way for more changes to follow, drastically leading to improved health. She decided to follow a vegan diet, ultimately cutting out processed foods, eating a diet primarily of fruits, vegetables, nuts and seed, legumes, and grains. Along with the diet changes, she decided exercise was needed to be added, so she did. In fact, she added a daily walking routine, getting at least 3,000 steps a day. Living in a one bedroom apartment can be a struggle, but not for this woman. To get in her daily steps, she began walking from her kitchen to the bedroom over and over again.
Now I realize this diet is different from a low carb diet, since it does incorporate grains and does not include meats, but the big point of cutting out the processed grains and processed foods is still there. Both diets have a core of vegetables and little to no processed foods. More importantly, this is how this woman was able to add exercise to her routine without going to a gym.