- #1

- 33

- 0

I have questions regarding this subject.

By definition, [tex]\limsup_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \sup_{n \geq k} f(x_n)[/tex] and [tex]\liminf_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \inf_{n \geq k} f(x_n)[/tex]. Say a sequence [tex]\{x_k\}[/tex] converging to [tex]0[/tex] from the left in the following example.

[tex]f(y) = \left\{

\begin{array}{ll}

y + 1 & \quad ,y > 0 \\

y & \quad ,y \leq 0

\end{array}

\right.[/tex]

Then [tex]\limsup_{k \to \infty} f(x_k) = \liminf_{k \to \infty} f(x_k) = f(0)[/tex].

Suppose we have another sequence [tex]\{x_k\}[/tex] converging to [tex]0[/tex] from the right. Then [tex]\limsup_{k \to \infty} f(x_k) = \liminf_{k \to \infty} f(x_k) > f(0)[/tex].

What is the difference between [tex]\limsup_{k \to \infty} f(x_k)[/tex] and [tex]\liminf_{k \to \infty} f(x_k)[/tex]? I don't see any difference.

By definition, [tex]\limsup_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \sup_{n \geq k} f(x_n)[/tex] and [tex]\liminf_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \inf_{n \geq k} f(x_n)[/tex]. Say a sequence [tex]\{x_k\}[/tex] converging to [tex]0[/tex] from the left in the following example.

[tex]f(y) = \left\{

\begin{array}{ll}

y + 1 & \quad ,y > 0 \\

y & \quad ,y \leq 0

\end{array}

\right.[/tex]

Then [tex]\limsup_{k \to \infty} f(x_k) = \liminf_{k \to \infty} f(x_k) = f(0)[/tex].

Suppose we have another sequence [tex]\{x_k\}[/tex] converging to [tex]0[/tex] from the right. Then [tex]\limsup_{k \to \infty} f(x_k) = \liminf_{k \to \infty} f(x_k) > f(0)[/tex].

What is the difference between [tex]\limsup_{k \to \infty} f(x_k)[/tex] and [tex]\liminf_{k \to \infty} f(x_k)[/tex]? I don't see any difference.

Last edited: