I hope you want to prove, If that is the case the best method is to use the definition of limit of a function.
I'm going to start by saying I wish I knew all the tags to make this look nice.
The question is prove the lim (1/n - 1/(n+1)) = 0
My sketch of this is ((1(n+1)-n)/n(n+1))=0 which leads to ((n+1-n)/n(n+1)) which leads to (1/n(n+1)) and finally (1/((n^2)+n)) which I wholeheartedly expect converges to 0.
Now how do I make this formal enough to convince someone else, or am I off the mark?