Why does the sequence 1, 1/2, 1/3, 1/4 approach zero?

The sequence $1,1/2,1/3,1/4,\dots$ approaches zero because it exhibits the property of convergence. In mathematics, a sequence converges to a limit if its terms can be made as close to that limit as desired by choosing a sufficiently large position in the sequence.12

For the sequence defined by $a_n=1/n$ (or $1/(n+1)$ depending on the starting index), every term is greater than zero, but the values decrease as $n$ increases. According to the formal definition established by Karl Weierstrass, for any chosen degree of closeness $ϵ>0$, one can find a point $N$ in the sequence such that all subsequent terms are within $ϵ$ of zero.2

Essentially, as $n$ tends toward infinity, the value of $1/n$ becomes arbitrarily small. While the terms never actually reach zero, they become indistinguishable from it for any practical degree of precision, confirming that zero is the limit of the sequence.12