Need help with 'Compact set Prove' question

[pantin]

Let L_i denote the line segment in R^2 between the points (-1/i, 1/i) and (1/i, 1/i) with i=1,2,3...
Is the union S=U_(i=1 to infinite)L_i compact? Justify.




after I drew the graph, I just noticed that this set is not closed, because all the start points and end points of those segment are not included, so why need to prove?

 

[HallsofIvy]

Let L_i denote the line segment in R^2 between the points (-1/i, 1/i) and (1/i, 1/i) with i=1,2,3...
Is the union S=U_(i=1 to infinite)L_i compact? Justify.




after I drew the graph, I just noticed that this set is not closed, because all the start points and end points of those segment are not included, so why need to prove?

None of those open intervals contains its endpoints certainly in that segment. You need to show that the UNION of all those intervals is an interval that does not include its endpoints. That is, I think, true and easy to prove.

 

[Moo Of Doom]

Where does it say the line segments don't include the endpoints? Are you sure they don't? If they don't, then S is clearly not closed, since the endpoints aren't in any of the L_i. But what if the line segments do include the endpoints?

 

[pantin]

oh yes! haha, you are smart, it doesn't say the line segments do not include endpoints,that's just the coordinate sign!

but then how to continue..

 

[Moo Of Doom]

A subset of R^2 is compact only if it's closed and bounded. S is clearly bounded, so we just need to check if it's closed. This means it contains all it's limit points.
You've drawn the graph. Does it look like there's a limit point that isn't in any of the intervals L_i?

 

[pantin]

i got it, it doesn't contain 0 , which is the limit of the sequence

thx a lot