On the beauty of convergence

Getting closer

A sequence converges when its terms eventually stay within any prescribed distance of a single value.

The definition

Định nghĩa 1. A sequence $(a_n)$ converges to $L$ if for every $\epsilon >0$ there exists $N$ such that $n\ge N$ implies $\mathrm{\mid }{a}_n-L\mathrm{\mid }<\epsilon$.

A familiar example

For $a_n=1\mathrm{/}n$, choose an integer $N>1\mathrm{/}\epsilon$. Then $n\ge N$ gives $\mathrm{\mid }{a}_n\mathrm{\mid }<\epsilon$.

Why precision matters

The order of quantifiers is the heart of the definition. We do not pick one accuracy for all purposes: the sequence must answer every challenge.