My weight is Normal. Can I lose 20lbs simply by eating properly and exercising at a gym?

Both my current weight (135) and goal weight (115) are within my normal BMI range, just so you know I'm not trying to be underweight. However I'd imagine trying to lose a full a full 20 lbs when your weight is already normal to begin with would be a tricky task. My friends tell me that I don't need a strict diet (contrarily to what I'd expect) and that if I simply eat properly ad exercise regularly that I will eventually reach my goal. They said that if I stopped this 950 calories/day diet I'm currently on and start eating properly, that they would pay to sign me up to a gym. It's a tempting offer, but I'm a little skeptical... will this method really work?
 
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