# How to prove that $\liminf$ or $\limsup$ exists

I have a Cauchy sequence $$x_n$$ and as part of my proof I am trying to prove that $$\liminf_{n \to \infty}x_n \leq \limsup_{n \to \infty}x_n$$. So far I have shown that $$\inf\{x_n : n \geq k\} \leq \sup\{x_n : n \geq k\}$$, but I'm not sure how to prove that these limits exist in the first place so I can make the comparison.

I'm thinking somehow showing that they are either decreasing or increasing, and because they are bounded (Cauchy sequence is bounded).

Any hints would be appreciated!

• Which limits are you having trouble proving exist? Oct 21, 2021 at 17:40
• @user6247850 proving the existence of either $\limsup$ or $\liminf$
– user843046
Oct 21, 2021 at 17:41
• I am guessing that the definition you are working with is $\liminf x_n = \lim_{k \rightarrow \infty} \inf\{x_n : n \ge k\}$, with $\limsup$ defined similarly. Then I agree you should show they are monotone. Specifically, what is the relation between $\inf\{x_n : n \ge k\}$ and $\inf \{x_n : n \ge k+1\}$? Oct 21, 2021 at 18:00

Hint: Let $$A$$ and $$B$$ be two non-empty bounded above subsets of reals. Suppose that $$A\subseteq B$$, then $$\sup(A)\leq\sup(B)$$.
Now, note that if $$k_1\leq k_2$$, then $$\{x_n:n\geq k_1\}\supseteq\{x_n:n\geq k_2\}$$. By using the above lemma, the monotonicity of the sequence $$(\sup\{x_n:n\geq k\})_{k\in\mathbb{N}}$$ follows. You can use similar argument to prove the monotonicity of the sequence $$(\inf\{x_n:n\geq k\})_{k\in\mathbb{N}}$$.