Question reagarding liminf definition

[transgalactic]

regarding this definition

<br /> <br /> a = \lim \bigg( \inf \{ a_k | k\geq n\} \bigg) <br />

the sequence
<br /> <br /> \inf \{ a_k | k\geq n \}<br />

is non decreasing . its inf gets bigger or not changing in each following sequence
so its limit is its least upper bound

am i correct??

 

[HallsofIvy]

Yes.

 

[transgalactic]

what is the relations between this liminf and
all the members in the sequence

is it bigger or smaller then all of them?

 

[HallsofIvy]

There is no requirement that a "liminf" be smaller or larger than all members of the sequence. The liminf of a sequence is the least upper bound of all subsequential limits.

For example, suppose a_n is (n-6)/2n if n is odd, -(n-6)/2n if n is even. Then {a_n} is {-5/2, 1, -1/2, 1/4, -1/10, 0, 1/14, -1/8, ...}. For n odd, we have a sequence that converges to 1/2. For n even, we have a sequence that converges to -1/2. The liminf is the smaller of those, -1/2 but there is a number in the sequence less than -1/2. The limsup is 1/2 but there is a term of the sequence larger than 1/2. We can change any finite number of terms in a sequence with changing any subsequential limits so there cannot be any relation between the limit or liminf or limsup and individual terms of the sequence.