How can a person actually stick to a diet?

If I were to answer your quetion in one word it would be "motivation".
But since there is room for more let me further elucidate. Any activity you take upon in your life has to be backed with the right amount of motivation to see it completed. Depending upon the intensity and difficulty of a task one requires higher amounts of motivation. Or else every single activity become boring and you tend to pull out of it sooner or later, whether it's getting a good physique, learning how to drive or trying to analyze the stock market.
In your particular situation however it's a little trickier than it may appear to be. Even though I get your points of a diet to be a science of calories in vs calories out, different people/bodies function differently. So while you're trying to cut down on calories and carbs (reducing carbs maynot be the best idea) you simultaneously have to mould your body to function in a way that'll help it to shape the way you'd want it to be and more importantly in a sustainable way for it to last you in the long run.
So, here are a few tips:
- Try to work on your metabolism. Increase your metabolism rate, do high amounts of cardio but do it in the right way. Wake up and go for high intensity jogs if not sprints, use the rowing machine, follow particular metabolic increaing work outs. Since you're still 21 this would be the ideal stage to push yourself to your limit, because not only are you at your prime age to mould your body the way you want, fixing your health now means eradicating future health problems.
- Follow a proper wholesome diet. Often people think starving themseleves is the best means to attain weight loss, but that's not a good idea. When you're striving for weight loss, ideally you should strive for a healthy body and life style and not just try to lose weight. Because losing weight doesn't not necessarily ensure a healthy body, but a healthy body/lifestyle will ensure you to lose weight till the right amount. Eat the right kind of fruits and veggies. Regulate your protien/carbs in take. Go low on the fat. Drink lots of water. One can actually research and figure out what kind of a diet they should ideally follow based on their body type.
- Keep the right motivation. The last and most important point would be about the fuel to keep yourself going in order to achieve the end goal. Often people start off well but tend to pull out eventually. This happens because they tend to forget the reason they had jumped into this quest in the first place. And if the goal is forgetten you're basically lost. So it's of prime importance to always keep in mind where you want to reach and what you want to achieve and how doing all this hard work will pay off eventually and get you that happiness you strive for. You could keep a photo of Cristiano Ronaldo in your wallet for motivation, or you could keep a photo of Scarlett Johanson too if you think getting a good physique will get you pretty women, whatever works for you. Always be positive, after a hard work out feel that you've burnt a good amount of calories and gained some muscle. Keep telling yourself in your head what you want and where you want to be. Drill it so hard in your sub conscience that the results will be inevitable. And work hard, with that I'm sure you'll not only have your health in place but pretty much your entire life.
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