- #1

cbarker1

Gold Member

MHB

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- Homework Statement
- Let ##S## be a subset of ##\mathbb{R}^2## with the standard metric. Show that if there exists a sequence ##(x_n, y_n)## in ##S## s.t. ##|(x_n,y_n)|\ge n## for all ##n \ge 1##, then ##S## is unbounded

- Relevant Equations
- A set ##S## is bounded if there is a closed ball B(r,p)=\{(x,y)\in \mathbb{R}^2: |(x,y)-p|\le R}\ such that ##B(r,p)## is a subset of ##S##.

Dear Everyone,

I am attempting a proof of contradiction for this problem. I am stuck on next step.

My attempt:

Assume that ##S## is bounded. Choose a ##N=\text{greatest integer function of} R+1##...

Here is where I am stuck. I want to show that the sequence is moving away from the closed ball as N is getting larger.

Thanks,

Cbarker1

I am attempting a proof of contradiction for this problem. I am stuck on next step.

My attempt:

Assume that ##S## is bounded. Choose a ##N=\text{greatest integer function of} R+1##...

Here is where I am stuck. I want to show that the sequence is moving away from the closed ball as N is getting larger.

Thanks,

Cbarker1

Last edited: