# Thread: lim sup and lim inf proofs

1. ## lim sup and lim inf proofs

Prove that if k > 0 and (s_n) is a bounded sequence, lim sup ks_n = k * lim sup s_n and what happens if k<0?

My attempt: If (s_n) is bounded, then lim sup s_n = S, where S is a real number, so lim sup (ks_n)= lim sup(k)(s_n)=S*lim sup(k) = S*k = k*lim sup s_n.

Let lim inf s_n = L. If k<0, then lim sup (ks_n) = -lim inf(-k)(s_n)= -L*lim inf (-k) = -k *- L = k * lim inf (s_n).

Is this correct?

2. Originally Posted by Pinkk
Prove that if k > 0 and (s_n) is a bounded sequence, lim sup ks_n = k * lim sup s_n and what happens if k<0?

My attempt: If (s_n) is bounded, then lim sup s_n = S, where S is a real number, so lim sup (ks_n)= lim sup(k)(s_n)=S*lim sup(k) = S*k = k*lim sup s_n.

Let lim inf s_n = L. If k<0, then lim sup (ks_n) = -lim inf(-k)(s_n)= -L*lim inf (-k) = -k *- L = k * lim inf (s_n).

Is this correct?
$\sup_{m\leqslant n}\{k\cdot a_n\}=k\sup_{m\leqslant n}a_m$

3. I'm not sure I follow. Was my proof incorrect?

4. Originally Posted by Pinkk
I'm not sure I follow. Was my proof incorrect?
We have that $\limsup\text{ }a_n=\lim_{n\to\infty}\{\sup_{n\leqslant m}a_m\}$ using the above see that $\limsup\text{ }k\cdot a_n=\lim_{n\to\infty}\{\sup_{m\leqslant n}k\cdot a_m\}=$ $\lim_{n\to\infty}\{k\cdot\{\sup_{m\leqslant m}a_m\}=k\lim_{n\to\infty}\{\sup_{m\leqslant n}a_m\}=k\limsup\text{ }a_n$. I actually can't follow what you did :S

5. I used the property that, given a sequence $(s_{n})$ that converges to $s$ and any other sequence $(t_{n})$, then $lim\, sup\, (s_{n}t_{n}) = s \cdot lim\, sup\, (t_{n})$. In my proof, I used the sequence $(k)$, which obvious converges to $k$.

6. Originally Posted by Pinkk
I used the property that, given a sequence $(s_{n})$that converges to $s$ and any other sequence $(t_{n})$, then $lim\, sup\, (s_{n}t_{n}) = s \dot lim\, sup\, (t_{n})$. In my proof, I used the sequence $(k)$, which obvious converges to $k$.
Well then that is obviously correct, if you have already proved that theorem. But, that is killing a fly with an elephant gun in my opinion.

7. I guess it isn't a very good exercise then, since the general theorem was proven in the chapter before the exercises.