Question

Cauchy’s general principle of convergence for sequences. To check whether the

sequence (1/2n)

is convergent or not.

Answer 1

Sequence $A_n$ is convergent if and only if for every $\varepsilon >0$  there is a natural number N such that $|A_n-A_{n+p}|<\varepsilon$ for all $n>N$ and natural number $p$

The sequence $A_n=1/2n$, then for each $\varepsilon$ let's take the nearest natural number $N\geqslant(1/2\varepsilon)$ , hence

take a look at $|A_n-A_{n+p}|: 0<A_{n+p}<A_n, \,then\, |A_n-A_{n+p}|<A_n,$

$|A_{n+p}-A_n|<A_n=1/2N<1/(2*1/\varepsilon)=\varepsilon/2<\varepsilon$

finally

$|A_{n+p}-A_n|<\varepsilon$. Thus, the sequence $A_n$ is convergent by Cauchy’s general principle of convergence.