Exercise 3.5

Question

Exercise 5:
    For any two real sequences $\{a_n\}$, $\{b_n\}$, prove that $$\limsup_{n\rightarrow\infty}(a_n+b_n)\le\limsup_{n\rightarrow\infty}a_n+\limsup_{n\rightarrow\infty}b_n,$$ provided
    that the sum on the right is not of the form $\infty-\infty$.

ghostofgarborg · Accepted Answer

If $\limsup a_n = \infty$ and $\limsup b_n > -\infty$, the result is trivial. 
    If $\limsup a_n = - \infty$ and $\limsup b_n < \infty$, then 
    $b_n$ is bounded above by $B$ and given a number $N$
    $$ \# \{ n : a_n + b_n > N \} \leq \# \{ n : a_n > N - B \} $$
    the latter of which is finite. Therefore $\limsup (a_n + b_n) = -\infty = \limsup a_n + \limsup b_n$.
    
    We can therefore assume that both $\limsup a_n$ and $\limsup b_n$ are finite.
    There is a subsequence such that $a_{n_k} + b_{n_k} \to \limsup (a_n + b_n)$.
    By choosing this subsequence right, i.e. if necessary passing to another subsequence, 
    we can make sure that $ a_{n_k} $ is convergent. Then $ b_{n_k} = (a_{n_k} + b_{n_k}) - a_{n_k}$
    is convergent, so 
    $$ \limsup_n (a_n + b_n) = \lim_k (a_{n_k} + b_{n_k}) = \lim_k a_{n_k} + \lim_k b_{n_k} \leq \limsup_n a_n + \limsup_n b_n $$
    which finishes the proof.

Exercise 3.5

Answers

Comments

Exercise 3.5

Answers

Comments

Add answer