A problem on limits

Question

When is it the case that $\limsup _{n}(\alpha a_{n})=\alpha \limsup _{n} a_{n} ?$ When does it not hold?

I believe that this holds when $\alpha \geq$ 0, then the equality can be obtained. Are there any other conditions?

Answer 1

This holds only if $\alpha \geq 0$, as the hint above mentions.

For $\alpha < 0$ this becomes false. For example,

$$\limsup_{n \to \infty} (-1) (-1)^n = 1$$ $$-\limsup_{n \to \infty} (-1)^n = -1$$