# Have I correctly proven $\limsup_{n \to \infty} x_n+\liminf_{n \to \infty} y_n \le \limsup_{n \to \infty} (x_n+y_n)$?

I want to prove: $$\limsup_{n \to \infty} x_n+\liminf_{n \to \infty} y_n \le \limsup_{n \to \infty} (x_n+y_n)$$. Is the proof I have written below correct? Have I overcomplicated the proof, overlooking a simpler method? Note that it is important, at my level of ability, to include and justify every step.

First note that $$\limsup_{n \to \infty} x_n=-\liminf_{n \to \infty} (-x_n)$$ follows directly from the reflection principle of $$\sup$$/$$\inf$$ and the definiton of $$\limsup$$ and $$\liminf$$.

$$\limsup_{n \to \infty} x_n+\liminf_{n \to \infty} y_n=$$

$$-\liminf_{n \to \infty} (-x_n)+\liminf_{n \to \infty} y_n=$$

$$-(\liminf_{n \to \infty} (-x_n)-\liminf_{n \to \infty} y_n)=$$

$$-(\lim_{n \to \infty} (\inf\{-x_k:k\ge n\}- \inf\{y_k:k\ge n\}))=$$

$$\lim_{n \to \infty} (-\inf\{-x_k:k\ge n\}+ \inf\{y_k:k\ge n\})=$$

$$\lim_{n \to \infty} (\sup\{x_k:k\ge n\}+ \inf\{y_k:k\ge n\})$$

but, $$\sup\{x_k:k\ge n\}+ \inf\{y_k:k\ge n\} \le \sup\{x_k:k\ge n\}+y_\alpha$$ for all $$\alpha \ge n$$

and because $$\sup\{x_k:k\ge n\}+y_\alpha=\sup\{x_k +y_\alpha:k\ge n\}$$ for all $$\alpha \ge n$$

we must have: $$\sup\{x_k:k\ge n\}+ \inf\{y_k:k\ge n\} \le \sup\{x_k +y_k:k\ge n\}$$ for all $$n$$

therefore $$\lim_{n \to \infty} (\sup\{x_k:k\ge n\}+ \inf\{y_k:k\ge n\}) \le$$ $$\lim_{n \to \infty} \sup\{x_k +y_k:k\ge n\}=\limsup_{n \to \infty} (x_n+y_n)$$.

I use the properties $$\sup(A) = -\inf(-A)$$ and $$\sup_{n\in I}(a_n+b_n) \leq \sup_{n\in I} a_n + \sup_{n\in I} b_n \text{ for } I\subseteq\mathbb{N}$$

You can then show for $$I\subseteq\mathbb{N}$$ \begin{align*} & \sup_{n\in I}(x_n) \\ & = \sup_{n\in I}(x_n+y_n - y_n) \\ & \leq \sup_{n\in I}(x_n+y_n) + \sup_{n\in I}(-y_n) \\ & = \sup_{n\in I}(x_n+y_n) - \inf_{n\in I}(y_n) \end{align*}

You can then use this to show the desired result by applying this to the sup/inf sequences of the limsup/liminf

• I am also looking for feedback on correctness of my proof.
– user796511
Sep 24, 2022 at 9:33

Everything looks good - good job! Since you asked whether you could simplify it further I tried to write out your proof with as few lines as possible and I included it below. Though, if your Prof is insisting every line be absolutely clear - then your proof might be a little better to submit.

LHS $$=\limsup _{n \rightarrow \infty} x_n+\liminf _{n \rightarrow \infty} y_n$$

$$\quad \quad =\lim_{n \to \infty} (-\inf\{-x_k:k\ge n\})+ \lim_{n \to \infty}(\inf\{y_k:k\ge n\})$$, from reflection and def'n

$$\quad \quad =\lim_{n \to \infty} (-\inf\{-x_k:k\ge n\}+ \inf\{y_k:k\ge n\})$$

$$\quad \quad =\lim_{n \to \infty} (\sup\{x_k:k\ge n\}+ \inf\{y_k:k\ge n\})$$

$$\quad \quad \leq \lim_{n \to \infty} (\sup_k\{x_k + y_\alpha :k\ge n, \forall\alpha\geq n\})$$

$$\quad \quad \leq \lim_{n \to \infty} (\sup\{x_k + y_k \})$$

$$\quad \quad = \limsup_{n \to \infty} (x_n+y_n)$$