0
votes
1answer
84 views

Proof that $\limsup (a_nb_n)=\limsup a_n \lim b_n$

Let $a_n$ be a bounded sequence and let $b_n$ be a convergent sequence. Prove that $\limsup⁡〖(a_n b_n )=\lim sup⁡〖a_n 〗 \lim⁡b_n 〗$. Here is what I got Proof Let $a_n$ be a bounded sequence and ...
5
votes
2answers
204 views

Limit superior inequalities proof

Let $a_n$ be a positive sequence. Prove that $$\limsup_{n\to \infty} \left(\frac{a_1+a_{n+1}}{a_n}\right)^n\geqslant e.$$
1
vote
3answers
257 views

Proof for limit superior's property: $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ [duplicate]

Let $a_n,b_n>0$ for all $n\in\mathbb N$. Prove that $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ I know that $\limsup (a_n+b_n ) \leq \limsup a_n + \limsup b_n$. But I don't know how ...
4
votes
2answers
200 views

limit superior and limit inferior proof

$$\limsup \left(\frac 1{a_n} \right)=\frac 1{\liminf(a_n )} $$ I know this is true base on the definition of $\limsup$ and $\liminf$, but I don't know how to prove it formally.
0
votes
1answer
631 views

Lim Sup and Lim Inf

This is a question with a few components to it. In each one, I give my attempt at a solution. Thank you in advance for any suggestions/answers/advice on how to solve this problem. For bounded ...
2
votes
1answer
647 views

lim sup inequality proof - is this the right way to think?

I have tried to read many proofs of this but I'm not sure I get it, so please bare with me. Show that $\lim_{n \rightarrow \infty} \sup (a_n+b_n) \leq \lim_{n \rightarrow \infty} \sup (a_n)+lim_{n ...
4
votes
1answer
454 views

limit superior of a sequence proof

Let $(x_{n})\in\mathbb{R}^{+}$ be bounded and let $x_{0}=\lim\sup_{n\rightarrow\infty}x_{n}$. $\forall\epsilon>0$, prove that there are infinitely many elements less than $x_{0}+\epsilon$ and ...