Question

Sequence function(n)=n/n+1

Answer 1

Prove that a sequence function(n)=n/n+1 converges to 1

Solution

$\lim\limits_{x\to \infin}\frac{n}{n+1}=1$

We need to show that for any given $\in \space \gt0,$  there is a natural number N such that if $n\gt N$, then

$|\frac{n}{n+1}-1|=|\frac{-1}{n+1}|\lt \in$

NB: $\frac{1}{n+1}\lt\frac{1}{n}$ for any $N\in \N$

for any given $\in$ by Archimedean property, there exists $N\in \N$ such that $\frac{1}{N}\lt\in.$

Therefore, if $n\gt N$ then, $\frac{1}{n+1}\lt\frac{1}{n}\lt\frac{1}{N}\lt \in$