Little to none of these claims are true or the improvement is so slight that it's not even realistic to worry about it. Working out raises your metabolism more than anything else (FYI raising your metabolism is not something that can be done quickly or by a magic pill. It takes months of body recomposition to raise your metabolism)
While adding a little lemon or lime juice to your water can be of benefit to one's kidneys and bladder, there is no proof that it will aid in weight reduction.
However, if you are more willing to reach your daily water intake when your water has been dressed up with a few ice cubes and thin slices of lemon or lime, then by all means keep up the practice!
Bottom line, the best way to reduce one's weight is to reduce caloric intake and increase activity. Aim for lasting lifestyle change. As dozens of us here can atest, there is no quick fix, magic bullet, or secret cure.
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