How to prove that the infimum of the sum of two sequences is at most the sum of their infima?

Question

I'm trying to show: $\inf(x_k+y_k) \leq \inf(x_k)+\inf(y_k)$. So I did:

Let $r>0$, there is a $k_1$ such that $x_{k_1}\leq \inf(x_k)+r/2$ and there is a $k_2$ such that $y_{k_2}\leq \inf(y_k)+r/2$.

Then: $x_{k_1}+y_{k_2} \leq \inf(x_k)+\inf(y_k)+r$

I'd like to take the inf and make $r \rightarrow 0$, but since $k_1$ and $k_2$ may be different I can't conclude. Please somebody help!

Answer 1

Let $x_k=(-1)^k$, $y_k=(-1)^{k+1}$. Then $x_k+y_k=0$ for all $k$, hence the $\inf$ is $0$ as well. However, $\inf x_k=\inf y_k=-1$.