Algebra of Limit Superior, Proof Verification of $\limsup (a_n + b_n) \leq \limsup (a_n) + \limsup (b_n)$ I was trying to prove the following inequality 
$$\limsup (a_n + b_n) \leq \limsup (a_n) + \limsup (b_n)$$
for the case when both are Real Numbers. 
Let $a_{{n}_k}$ and $b_{{n}_k}$ be subsequences converging to limits $a_1$ and $b_1$, then given any $\epsilon > 0$, there exists $N_1$ and $N_2$ such that 
$| a_{{n}_k} - a_1| < \epsilon /2\qquad\forall k>N_1$, and 
$| b_{{n}_k} - b_1| < \epsilon /2\qquad\forall k>N_2$
and by triangle inequality we have 
$$| a_{{n}_k}  + b_{{n}_k} - (a_1+b_1) | < \epsilon\qquad\forall k>\max\{N_1,N_2\}$$
And since we know there exists subsequences converging to $\limsup a_n$ and $\limsup b_n$, we have $(a_{{n}_k} + b_{{n}_k}) \rightarrow\limsup a_n+\limsup b_n$. Something seems to be wrong, can anyone help me with this, Thank you.  
 A: First. This is true, provided that $\limsup a_n$ and $\limsup b_n$ can be added, i.e., the case that one of them is $\infty$ and the other is $-\infty$ should be excluded.
Second. If $\limsup a_n=\infty$ or $\limsup b_n=\infty$, then there is nothing to prove, as the left hand side is equal to $\infty$.
Third. If $\limsup\, (a_n+b_n)=-\infty$, there is also nothing to prove. 
So, assuming that both sequences are bounded, we pick a subsequence
$$
a_{n_k}+b_{n_k}\to\limsup\, (a_n+b_n).
$$
As both $a_{n_k},b_{n_k}$ are bounded, we may pick a common sub-sub-sequence $n_\ell$, such that both $a_{n_\ell},b_{n_\ell}$ converge. Say
$$
a_{n_\ell}\to a\le \limsup\, a_n, \qquad
b_{n_\ell}\to b\le\limsup\, b_n
$$
and hence
$$
\limsup\, a_n+\limsup\, b_n \ge a+b=\lim\, (a_{n_\ell}+b_{n_\ell})=\lim\, (a_{n_k}+b_{n_k})=\limsup\, (a_n+b_n)
$$
A: The problem in your proof, as mentioned by @Lee Chun Min, is that you claimed that there was a subsequence of $c_n=a_n+b_n$ converging to $\limsup a_n+\limsup b_n$ from the fact that there exists subsequences $a_{n_k}$ and $b_{N_k}$ converging to $\limsup a_n$ and $\limsup b_n$ respectively. This is not true until $$N_k=n_k,\forall k\in\mathbb{N}$$
So, in short, your proof is incomplete. This is exactly what @Lee Chun Min pointed out by using the term 'indices'.
A: Hint: Assuming $a_n$ and $b_n$ are bounded below ! 
Now let $a_m + b_m \to \delta =\limsup (a_n + b_n)$, and assume $a_{m_k}$ is a convergent subsequence of $a_m$ in $\Bbb R \cup \{ \infty \}$. WLOG assume $b_{m_k}$ is convergent too (other wise you can extract convergent subsequence from both $a_{m_k}$ and $b_{m_k}$). Now Observe that $a_{m_k} + b_{m_k} \to \delta$. DONE
