Show by use of an example, that: (note the strict inequality)$$\lim_{n \rightarrow \infty} \sup (x _{n}+y _{n}) \lim_{n \rightarrow \infty} \inf (x _{n})+\lim_{n \rightarrow \infty} \inf (y _{n}) $$

Question

Show by use of an example, that: (note the strict inequality)$$\lim_{n \rightarrow \infty} \sup (x _{n}+y _{n}) < \lim_{n \rightarrow \infty} \sup (x _{n})+\lim_{n \rightarrow \infty} \sup (y _{n}) $$ and also
$$\lim_{n \rightarrow \infty} \inf (x _{n}+y _{n}) > \lim_{n \rightarrow \infty} \inf (x _{n})+\lim_{n \rightarrow \infty} \inf (y _{n}) $$

anonymous · Answer

Where $$x _{n}\ and  \y _{n}$$ are bounded sequences.

anonymous · Answer

Take
$$
x_n=(-1)^n\
y_n=(-1)^{n+1}\
x_n+y_n=0\
\lim \sup  x_n =1\
\lim \sup y_n =1\
\lim \sup (x_n+ y_n)=0\
0 < 2\

\lim \inf  x_n =-1\
\lim \inf y_n =-1\
\lim \inf (x_n+ y_n)=0\

-2 <0

$$