Answer to Question #122936 in Real Analysis for Ruksan

Question

Let {vn} be a sequence in R2
, say vn = (xn, yn). Give R2
the || ||∞
norm. Show that limn→∞ vn → v if and only if limn→∞ xn = x and
limn→∞ yn = y where v = (x, y).

Answer 1

Given that $\{ v_n\}$ be a sequence in $\R^2$ .

Let $\{ v_n\}=(x_n,y_n)$ and $v=(x,y)$ .

Let $lim_{n\to \infty} v_n=v$

Claim : $x_n\rightarrow x \ and \ y_n\rightarrow y \ as \ n\rightarrow \infty$

i,e $lim_{n\to \infty} x_n=x \ and \ lim_{n\to \infty} y_n=y$ .

As $lim_{n\to \infty} v_n=v$ ,so for a given $\epsilon >0$ there exist a $K\in \N$

such that $||v_n-v||_{\infty}<\epsilon \ \forall n \geq K$

$\implies ||x_n-x,y_n-y||_{\infty}<\epsilon \ \forall n \geq K$

$\implies max\{ |x_n-x|,|y_n-y|\}<\epsilon$ $\forall n \geq K$

$\implies |x_n-x|<\epsilon \ and \ |y_n-y|<\epsilon \$ $\forall n\geq K$

Hence $lim_{ n\to \infty} x_n=x \ and \ lim_{n\to \infty} y_n=y.$

Conversely , assume that $lim_{n\to \infty}x_n=x \ and \ lim_{ n\to \infty}y_n=y$

Claim: $lim_{n\to \infty} v_n=v$ .

Therefore , for any $\epsilon >0 , \exist \ N_1 , N_2$ such that

$| x_n-x|<\epsilon$ for all $n\geq N_1$

and $|y_n-y|<\epsilon$ for all $n\geq N_2$ .

Now , $||v_n-v||_{\infty}=||x_n-x,y_n-y||_{\infty}$

$=Max \{ |x_n-x|,|y_n-y|\} <\epsilon$ $\forall n\geq N$

Where $N=Max\{ N_1,N_2\}$ .

Hence , $lim_{n\to \infty}v_n=v.$