Is This Sequence Compact? Analysis of {1, 1/2, 2/3, 3/4, 4/5...}

[cragar]

Homework Statement

{1,1/2,2/3,3/4,4/5...} is this set compact.

The Attempt at a Solution

I think this set is compact because it contains its cluster point which is 1.
is this correct?

 

[micromass]

That is correct.

 

[HallsofIvy]

You could prove that by appealing to "any closed and bounded subset of the real numbers is compact" or directly from the definition:
Let $\{U_\alpha\}$, for [itex]\alpha[itex] in some index set, be an open cover for this set. Since 1 is in the set, 1 is in at least one of those- relabel the sets, if necessary, so that set is called $U_1$. Since $U_1$ is open, there exist a point p, in that set, and a number $\delta> 0$ such that $N_\delta(p)= \{x | |x- p|<\delta\}$ is in $U_1$. It then follows that if $n> 1/\delta$ $|1- (n-1)/n|= 1/n< \delta$ so that all except a finite number of the fractions in the sequence are in $U_1$. Every number in the sequence, $(n-1)/n$ for n less than that might be in as separate set in the original sequence but there are only a finite number of them.