show that $\inf \{x_n+y_n:n>N\}\ge \inf \{x_n:n>N\}+\inf \{y_n:n>N\}$

Question

show that $\inf \{x_n+y_n:n>N\}\ge \inf \{x_n:n>N\}+\inf \{y_n:n>N\}$.

I am stuck on proving this. I understand it intuitively. My approach is to first do the right part. By defining the infs of yn and xn. But don't see how to move forward. Thanks

Answer 1

Suppose first that $\inf\{y_n:n>N\}=L$, a real number. Then, for each $n$, we have $x_n+y_n\ge x_n+L$, and hence $$\inf\{x_n+y_n:n>N\}\ge \inf\{x_n+L:n>N\}=\inf\{x_n:n>N\}+L=\inf\{x_n:n>N\}+\inf\{y_n:n>N\}$$

On the other hand, if $\inf\{y_n:n>N\}=-\infty$, then the RHS of your inequality might not be well-defined, since it involves adding a real number to $-\infty$. If you extend $\mathbb{R}$ to include $-\infty$, then the RHS will always be $-\infty$, so the inequality is always true.