Can some explain the

lim supandlim inf?

In my text book the definition of these two is this.Let (sn) be a sequence in R. We define

lim

and

\lim\inf\ s_n = \lim_{N\rightarrow \infty}\inf\{s_n:n>N\}The right side of these two equality, can I think \sup\{s_n:n>N\} and \inf\{s_n:n>N\} as a sequence after n>N?

And how these two behave as n increases? My professor said that these two get smaller as n increases.

**Answer**

Consider this example:

3-\frac12,\quad 5+\frac13,\quad 3-\frac14,\quad 5+\frac15,\quad 3-\frac16,\quad 5+\frac17,\quad 3-\frac18,\quad 5+\frac19,\quad\ldots\ldots

It alternates between something approaching 3 from below and something approaching 5 from above. The lim inf is 3 and the lim sup is 5.

The inf of the whole sequence is 3-\frac12.

If you throw away the first term or the first two terms, the inf of what’s left is 3-\frac14.

If you throw away all the terms up to that one and the one after it, the inf of what’s left is 3-\frac16.

If you throw away all the terms up to that one and the one after it, the inf of what’s left is 3-\frac18.

If you throw away all the terms up to that one and the one after it, the inf of what’s left is 3-\frac1{10}.

. . . and so on. You see that these **inf**s are getting bigger.

If you look at the sequence of **inf**s, their **sup** is 3.

Thus the lim inf is the **sup** of the sequence of **inf**s of all tail-ends of the sequence. In mathematical notation,

\begin{align}

\liminf_{n\to\infty} a_n & = \sup_{n=1,2,3,\ldots} \inf_{m=n,n+1,n+2,\ldots} a_m \\[12pt]

& = \sup_{n=1,2,3,\ldots} \inf\left\{ a_n, a_{n+1}, a_{n+2}, a_{n+3},\ldots \right\} \\[12pt]

& = \sup\left\{ \inf\left\{ a_n, a_{n+1}, a_{n+2}, a_{n+3},\ldots \right\} : n=1,2,3,\ldots \right\} \\[12pt]

& = \sup\left\{ \inf\{ a_m : m\ge n\} : n=1,2,3,\ldots \right\}.

\end{align}

Just as the lim inf is a **sup** of **inf**s, so the lim sup is and **inf** of **sup**s.

One can also say that L=\liminf\limits_{n\to\infty} a_n precisely if for all \varepsilon>0, no matter how small, there exists an index N so large that for all n\ge N, a_n>L-\varepsilon, and L is the largest number for which this holds.

**Attribution***Source : Link , Question Author : eChung00 , Answer Author : Michael Hardy*