Answer to Question #329494 in Real Analysis for Sakshi Naik Gaonka

Question

Given a sequence ((xn,yn)) is R2 .prove that if ((xn,yn)) is bounded ,then (xn) and (yn) are bounded.

Answer 1

ANSWER

By the definition, the sequence $\left\{ \left( { x }_{ n }{ ,y }_{ n } \right) \right\}$ is bounded in $R^2$ if there exists $M>0$ such that

$\left\| \left( { x }_{ n },{ y }_{ n } \right) \right\| =\sqrt { { \left( { x }_{ n } \right) }^{ 2 }+{ \left( { y }_{ n } \right) }^{ 2 } } \le M$ for all $n\in N$ .

Therefore

$\left| { x }_{ n } \right| =\sqrt { { \left( { x }_{ n } \right) }^{ 2 }\quad } \le \sqrt { { \left( { x }_{ n } \right) }^{ 2 }+{ \left( { y }_{ n } \right) }^{ 2 } } \le M$ for all $n\in N$ .

and

$\left| { y }_{ n } \right| =\sqrt { { \left( { y }_{ n } \right) }^{ 2 }\quad } \le \sqrt { { \left( { x }_{ n } \right) }^{ 2 }+{ \left( { y }_{ n } \right) }^{ 2 } } \le M$

Equivalent to

$-M\leq x_{n}\leq M$ . for all $n\in N$

and

$-M\leq y_{n}\leq M$ for all $n\in N$

So, the sequences $\left\{ \left( { x }_{ n } \right) \right\} ,$ $\left\{ \left( {y }_{ n } \right) \right\}$ are bounded in $R$ .