Answer to Question #330889 in Real Analysis for Sakshi Naik Gaonka

Question

Let ((Xn, Yn)) and ((Un, Vn)) be sequences in R2, and let

(X0, Y0), (U0, Vo) belong to R2.

(i) If (Xn,Yn) converges to (X0, Y0) and (Un, Vn) converges to (Uo, Vo), then (Xn,Yn)+(Un, Vn) converges to (X0,Y0) + (U0,V0) and (Xn,Yn).(Un,Vn) converges to (Xo,Y0)(Uo, V0).

ii) If (Xn, Yn) converges to (X0, Y0), then for any r belonging to R, r(Xn,Yn) converges to r(X0,Y0).

Answer 1

ANSWER

To prove the statements , we use the following propositions.

Proposition 1. Let $\mathbf{x_ {n}}=\left (x _{n} ,y_{n} \right )$ is sequence in $\R^{2}$ , $\mathbf{x }=\left (x ,y \right )$ .

$\mathbf{x_ {n}}\rightarrow \mathbf{x }(||\mathbf{x_ {n}}-\mathbf{x }||\rightarrow 0)$ if and only if $x_{n}\rightarrow x$ and $y_{n}\rightarrow y$.

Proposition 2.

a) If the sequence $(x_{n})$ converges to $x$ in $\R$ and $a\in\R$ , then the sequence $(ax_{n})$ converges to $ax$ ($\lim_{n\rightarrow\infty}a\cdot x_{n}=a\cdot \lim_{n\rightarrow\infty}x_{n}$ )

b) If the sequence $(x_{n})$ converges to $x$ in $\R$ and $(y_{n})$ converges to $y$ in $\R$ , then the sequence $( x_{n}+y_{n})$ converges to $x+y$ , the sequence $( x_{n}\cdot y_{n})$ converges to $x\cdot y$ $(\lim_{n\rightarrow\infty} ( x_{n}+y_{n})= \lim_{n\rightarrow\infty}x_{n}+\lim_{n\rightarrow\infty}y_{n},$

$\lim_{n\rightarrow\infty} ( x_{n}\cdot y_{n})=( \lim_{n\rightarrow\infty}x_{n})\cdot (\lim_{n\rightarrow\infty}y_{n}))$ .

(i) Since

$(x_{n},y_{n})+(u_{n},v_{n})=(x_{n}+u_{n},y_{n}+v_{n})$ and ,by the Proposition 1, $x_{n} \rightarrow x_{0}, y_{n}\rightarrow y_{0}, u_{n}\rightarrow u_{0}, v_{n}\rightarrow v_{0}$ , then ( by the Proposition 2 b)) $x_{n}+u_{n} \rightarrow x_{0}+u_{0}, y_{n}+v_{n}\rightarrow y_{0} +v_{0}$ . Hence (Proposition 1), $(x_{n}+u_{n},y_{n}+v_{n})\rightarrow (x_{0}+u_{0},y_{0}+v_{0})=(x_{0},y_{0})+(u_{0},v_{0})$ .

Therefore $(x_{n},y_{n})+(u_{n},v_{n})\rightarrow (x_{0},y_{0})+(u_{0},v_{0}).$

Since,

$(x_{n},y_{n})\cdot(u_{n},v_{n})= x_{n}\cdot u_{n}+y_{n}\cdot v_{n}$ and $x_{n}\cdot u_{n} \rightarrow x_{0}\cdot u_{0}, y_{n}\cdot v_{n}\rightarrow y_{0} \cdot v_{0}$ , then

$x_{n}\cdot u_{n}+y_{n}\cdot v_{n} \rightarrow x_{0}\cdot u_{0}+y_{0}\cdot v_{0} =$ $(x_{0},y_{0})\cdot(u_{0},v_{0})$ .

Or $(x_{n},y_{n})\cdot(u_{n},v_{n})$ converges to $(x_{0},y_{0})\cdot(u_{0},v_{0})$

(ii)$r\cdot (x_{n},y_{n})=(rx_{n},ry_{n})$ . By the Propositions 2a ),1), $\lim_{n\rightarrow\infty}r\cdot x_{n}=r\cdot x,$ $\lim_{n\rightarrow\infty}r\cdot y_{n}=r\cdot y,$ $(rx_{n},ry_{n})\rightarrow(rx_{0},ry_{0})=r( x_{0}, y_{0}).$

Or $r\cdot (x_{n},y_{n})\rightarrow$ $r( x_{0}, y_{0}).$