Can someone clearly explain about the lim sup and lim inf?

Can some explain the lim sup and lim 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 infs are getting bigger.

If you look at the sequence of infs, their sup is 3.

Thus the lim inf is the sup of the sequence of infs 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 infs, so the lim sup is and inf of sups.

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

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