et {${x_{n}}$} be a sequence and $x \in R$ such that there exists a $k \in N$ such that for all $n \ge k$, $x_n = x$.

Question

Let {${x_{n}}$} be a sequence and $x \in R$ such that there exists a $k \in N$ such that for all $n \ge k$, $x_n = x$. Prove that {${x_n}$} converges to $x$.

I tried to do $\lim x_{n} = x$ and $\mid x_{n} - x\mid = -x_{n} + x < -x_{k} +x < \epsilon$

But it doesnt make sense... To do what I did, I have to know that $x$ is supremum which is greater than or equal to $x_{n}$

Answer 1

You want to show that $$ \forall \epsilon>0,\exists N,\forall n:n\geq N\implies |x_n-x|<\epsilon $$

Let $\epsilon>0$. Taking $N:=k$ where $k$ is the number given in the problem statement, we get $$ \forall n:n\geq N\implies |x_n-x|=0<\epsilon $$ Done.