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Suppose $(a_n)^{n\to\infty}_{n=1}$ is a bounded sequence of real numbers. Prove that:

$$\liminf a_n \leq\limsup a_n$$

This makes sense as $\inf a_n$ is the lowest bound of $a_n$ and $\sup a_n$ is the lowest upper bound of $a_n$ and if $a_n$ converges $\liminf a_n =\limsup a_n = \lim a_n$.

However, I am struggling to prove this. Could I do this by contradiction?

Suppose $$\liminf a_n >\limsup a_n$$

$\liminf a_n$ is the limit of $\inf \{a_k:k\ge n\}$ and is the lowest bound of $a_n$ and $\limsup a_n$ is the limit of $\sup \{a_k:k\ge n\}$ and is the lowest upper bound of $a_n$. By definition, if $\liminf a_n >\limsup a_n$, then $\liminf a_n$ is obviously not the lowest bound. Contradiction.

Is this a way to prove this?

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3 Answers 3

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Hint: You could simply note that $\inf(S)\le \sup(S)$ for any non-empty set of real numbers $S$ (make sure you can show this!), and inequalities are preserved under taking limits.

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Let $(a_n)_{n = 1}^\infty$ be a bounded sequence of real numbers. That is, there is some real number $M$ such that $\vert a_n \vert \leq M$ for every $n \geq 1.$ Consider \begin{align*} \liminf_{n \to \infty} a_n &= \lim_{k \to \infty} (\inf_{n \geq k} a_n) \text{ and}\\ \limsup_{n \to \infty} a_n &= \lim_{k \to \infty}(\sup_{n \geq k} a_n). \end{align*} Let $k$ be a positive integer. If we can show that $\inf_{n \geq k} a_n \leq \sup_{n \geq k} a_n$, then the statement is proven. Observe that \begin{align*} \inf_{n \geq k} a_n &= \inf\{ a_n : n \geq k \} \leq a_k \leq \sup\{ a_n : n \geq k \} = \sup_{n \geq k} a_n. \end{align*} Since this holds for all $k \geq 1$, we have $\liminf_{n \to \infty} a_n \leq \limsup_{n \to \infty} a_n$ and we are done. $\blacksquare$

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Hint : You can (show and) use the following equivalent definitions of $\liminf$ and $\limsup$ : the $\liminf$ is the smallest limit point, and the $\limsup$ is the greatest limit point.

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  • $\begingroup$ Hi, thanks for the help, but I am more interested in whether what I have written is a correct/valid proof. $\endgroup$ Mar 26, 2019 at 1:01

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