Example of $\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$

Question

I am looking for an example demonstrating that $\lim\inf x_n+\lim \inf y_n<\lim \inf(x_n+y_n)$ but for the life of me i can't find one. any suggestions?

Answer 1

Let $(x_{n})_{n}$ such that $x_{2n} = 1$ and $x_{2n - 1} = -1$

Let $(y_{n})_{n}$ such that $y_{2n} = -1$ and $y_{2n -1 } = 1$

Then $\liminf x_{n} = -1$, $\liminf y_{n} = -1$

And $\liminf (x_{n} + y_{n}) = 0$

Answer 2

Try some alternating sequences where $\liminf x_n = \liminf y_n = -1$ but $x_n + y_n = 0$ for each $n$.

Answer 3

Choose $(x_n) = (1,-1,1,-1,\dots)$ and $(y_n) = (-1,1,-1/2,1,-1/3,\dots)$. Then $(x_n+y_n) = (0,0,1/2,0,2/3,\dots)$. This means $\liminf x_n = -1$, $\liminf y_n = 0$ and $\liminf (x_n + y_n)=0$. With this sequence one can also see that

\begin{equation} \limsup (x_n+y_x) < \limsup x_n + \limsup y_n. \end{equation}