Lim sups and lim infs homework

[koshn]

Homework Statement

http://i.imgur.com/ARy65.png

Homework Equations

Just general definitions of lim sups and lim infs

The Attempt at a Solution

I'm stuck on the part a. I realize that since a_n has a limit that means it is bounded and monotone. By I don't know how that implies that b_m is nonincreasing. The way I understand b_m is that b_1 = sup{a_m} and b_2 = sup{a_m,a_(m+1)} so is that correct? If so doesn't it depend on whether or not a_m is less than or greater than a_(m+1) and so on to know that b_m is nonincreasing?

 

[spamiam]

I'm stuck on the part a. I realize that since a_n has a limit that means it is bounded and monotone. By I don't know how that implies that b_m is nonincreasing. The way I understand b_m is that b_1 = sup{a_m} and b_2 = sup{a_m,a_(m+1)} so is that correct? If so doesn't it depend on whether or not a_m is less than or greater than a_(m+1) and so on to know that b_m is nonincreasing?

No, a_n is not assumed to converge (unless I missed something). The limsup is defined for any sequence, even divergent ones (although then sometimes the limsup may be infinity).

I don't think you totally understand the b_m. The way b_m is defined would give

$$b_1 = \sup\{a_1, a_2, a_3, a_4 \ldots\}$$
$$b_2 = \sup\{a_2, a_3, a_4 \ldots\}$$
$$b_3 = \sup\{a_3, a_4 \ldots\}$$

,etc. Basically, as m increases, the first m terms of (a_n) are omitted when taking the sup. Now take another look at the hint. Does that help?

 

[jbunniii]

koshn - What definition of lim sup are you using? I ask because part (d) is often given as the definition of lim sup, whereas you are being asked to prove it.

 

[koshn]

No, a_n is not assumed to converge (unless I missed something). The limsup is defined for any sequence, even divergent ones (although then sometimes the limsup may be infinity).

I don't think you totally understand the b_m. The way b_m is defined would give

$$b_1 = \sup\{a_1, a_2, a_3, a_4 \ldots\}$$
$$b_2 = \sup\{a_2, a_3, a_4 \ldots\}$$
$$b_3 = \sup\{a_3, a_4 \ldots\}$$

,etc. Basically, as m increases, the first m terms of (a_n) are omitted when taking the sup. Now take another look at the hint. Does that help?

Thanks that helped to clear the confusion I was having.

koshn - What definition of lim sup are you using? I ask because part (d) is often given as the definition of lim sup, whereas you are being asked to prove it.

The definition I'm using is that lim sup x_n is defined to be the infimum of all numbers b with the following property: There is an integer N so that x_n < b for all n >= N.