Finding two sequences with a limsup value Find two sequences 
  ⟨
  
    X
    n
  
  ⟩
 and 
  ⟨
  
    Y
    n
  
  ⟩
  such that limsup(Xn + Yn )=E=Euler's constant and limsup Xn + limsup Yn =Pi=π .
I couldn't think of a way to approach this problem. Can anyone help ?
 A: 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?) 
A: 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 $\limsup x_n+y_n=E$
A: 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 $$ \lim \sup X_n  + \lim \sup Y_n = \sup \{ 0 \; ,  e - \pi\} \; +  \sup \{ e \;, \pi \; \}= 0 +  \pi = \pi$$
