We want to prove that

To this end we have to show that we can win the game from the definition.

Assume we were given a positive ε. We need to find a natural number N so that for all n ≥ N we have |a_n − 1| < ε, that is,

|(n+1)/n − 1| < ε.

But this is an inequality and we can try to find out for which positive integers n it works:

We could drop the absolute value since for positive n, the fraction is also positive. We obtained a condition describing when the required inequality works, and fortunately for us, it is an inequality of the type "it works if n is large enough". In other words, it exactly fits the situation.

Now we can find the cutoff point N: We choose for it the first natural number that comes after 1/ε. Did we win the game?

We found a cutoff points. Is it the right one? If n ≥ N, then from N > 1/ε we also have n > 1/ε. Since the calculations above worked both ways, consequently we also for those n have |a_n − 1| < ε, exactly as needed.