Let $a_n$ and $b_n$ be bounded monotone sequences. What can we say about the monotonicity and convergence of the sum $a_n +b_n$?
By the monotone convergence theorem both sequences $a_n$ and $b_n$ are convergent so suppose that $a_n$ converges to $a$ and $b_n$ converges to $b$. Then for large enough $n$ we have that $a_n + b_n$ converges to $a+b$ so the sequence $a_n +b_n$ converges?
Now to talk about the monotonicity of the sum $a_n + b_n$ I assume we need to make some assumptions on $a_n$ and $b_n$. Suppose that $a_n \le a_{n+1}$ and $b_n \le b_{n+1}$ for every $n$. These sequences increasing and since they're bounded we have that $|a_n|\le M$ for every $n$ and $|b_n| \le N$ for every $n$. Summing we get that $a_n + b_n \le a_{n+1} +b_{n+1}$ so the sequence $a_n + b_n$ is increasing also and thus monotone and since both are bounded we have that $|a_n+b_n| \le M+N$ for every $n$?
Similar reasoning can be made if the sequences are decreasing, but the interesting part comes when say $a_n$ is increasing and $b_n$ is decreasing. My assumption is that the sum isn't monotone but it still would be bounded?
Best Answer
$(a_n)$ converges to $a$ and $(b_n)$ converges to $b$. The sum of two convergent sequences is convergent and converges to the sum of the limits, i.e. $(a_n + b_n)$ converges to $a+b$. (The statement “For large enough $n$ we have that $a_n+b_n$ converges ...“ makes no sense, though.)
Every convergent sequence is bounded, therefore $(a_n+b_n)$ is bounded. The monotony of the sum is not needed here.
You are right that if both sequences are increasing then the sum is increasing, and if both sequences are decreasing then the sum is decreasing.
If $(a_n)$ and $(b_n)$ have different monotony (one increasing and the other decreasing) then nothing can be said about monotony of the sum: It may be increasing, decreasing, or neither.
In fact any sequence can be written as the sum of an increasing and a decreasing sequence.