Finding two sequences with a limsup value

Question

Find two sequences ⟨ X n ⟩ and ⟨ Y n ⟩ such that limsup(X_n + Y_n )=E=Euler's constant and limsup X_n + limsup Y_n =Pi=π .

I couldn't think of a way to approach this problem. Can anyone help ?

Answer 1

Take $$ X_n = (0, \; e - \pi \; , 0 \; , e - \pi \; , 0 \; , e - \pi \; , 0 \; , e - \pi \; , ... )$$

$$ Y_n = ( e \; , \pi \; , e \; , \pi \; , e \; , \pi \; , e \; , \pi \; , ... ) $$

Then, $$ (X_n + Y_n) = ( e, e, e, ... ) \implies \limsup (X_n + Y_n) = \lim (X_n + Y_n) = e$$

and $$ \lim \sup X_n + \lim \sup Y_n = \sup \{ 0 \; , e - \pi\} \; + \sup \{ e \;, \pi \; \}= 0 + \pi = \pi$$

Answer 2

The sequences $x_n=(0,1,0,1,0,1,\dots)$ and $y_n=(1.0,1,0,1,0,\dots)$ are an example of sequences such that $$ \begin{align} \limsup{x_n+y_n}&=1\\ \limsup x_n+\limsup y_n&=2 \end{align}$$ Can you modify them to get sequences required in your post?

(Hint: What happens with limit superior if you replace $(x_n)$ by $(x_n+d)$ for some constant $d$? What happens if you replace $(x_n)$ by $(cx_n)$ for some constant $c>0$? In the other words, can you compute $\limsup c(x_n+y_n$ and $\limsup cx_n+\limsup cy_n$? Similarly, what is $\limsup (x_n+d+y_n)$ and $\limsup (x_n+d)+\limsup(y_n+d)$ equal to?)

Answer 3

Define $x_n=E-\pi$ if $n$ is odd, and $0$ if $n$ is even; $y_n=\pi$ if $n$ is odd, and $-\pi$ if $n$ is even.

Note $E<\pi$, then $\limsup x_n=0, \limsup y_n=\pi$. Hence $\limsup x_n+\limsup y_n=\pi$.

While $x_n+y_n=E$ if $n$ is odd, $-\pi$ if $n$ is even, hence $\limsup x_n+y_n=E$