Prove that: $\limsup_{n \rightarrow \infty} (a_n+b_n) \le \limsup_{n\rightarrow\infty}(a_n) + \limsup_{n\rightarrow \infty} (b_n)$

Question

I am reading Rudin's "Principles of Mathematical Analysis", and the following problem appears in it:

Problem 3.5:

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 the sum on the right is not of the form $\infty – \infty$

The solution in the manual goes as following:

Since the case when $\limsup_{n\rightarrow\infty}(a_n) = +\infty$ and $\limsup_{n\rightarrow\infty}(b_n) = -\infty$ has been excluded from consideration , we note that the inequality is obvious if $\limsup_{n\rightarrow\infty}(a_n) = +\infty$. Hence we shall assume that $\{a_n\}$ is bounded above.

Let $\{n_k\}$ be a subsequence of the positive integers such that $\lim_{k\rightarrow\infty}(a_{n_k} + b_{n_k}) = \limsup_{n\rightarrow\infty}(a_n + b_n)$.

Then choose a subsequence of the positive integers $\{k_m\}$ such that

$$\lim_{m\rightarrow\infty}(a_{n_{k_m}}) = \limsup_{k\rightarrow\infty}(a_{n_k})$$

The subsequence $a_{n_{k_m}} + b_{n_{k_m}}$ still converges to the same limit as $a_{n_k} + b_{n_k}$.

And proof continue.
I understand all the proof except for the last sentence, precisely:

The subsequence $a_{n_{k_m}} + b_{n_{k_m}}$ still converges to the same limit as $a_{n_k} + b_{n_k}$.

Why this is true?

I was thinking that we can also assume that $\{b_n\}$ also bounded so that it contains a series that converges and so $\limsup_{n\rightarrow \infty} (b_n)$ is finite but I don't know if this is useful and I couldn't go further.

Answer 1

After the answer from Kavi, and some extra effort I see that we can solve the problem as follows:

If $\limsup_{n\rightarrow\infty}(b_n) = +\infty$ then the inequality is also obvious so we can also (as we did with $\{a_n\}$) assume that $\{b_n\}$ is bounded.

Since both sequences are bounded, $\{a_n+b_n\}$ also bounded, and there is a subsequence of $\{a_n+b_n\}$ that converges so $\limsup_{n\rightarrow\infty}(a_n+b_n)$ is finite and since: $$\lim_{k\rightarrow\infty}(a_{n_k} + b_{n_k}) = \limsup_{n\rightarrow\infty}(a_n + b_n)$$

Then $\lim_{k\rightarrow\infty}(a_{n_k} + b_{n_k})$ is also finite and therefore converges.

Then every subsequence of it also converges to the same limit.

And this answers my question, but please correct me if there is a flaw somewhere.