Prove that if $\lim_{n\to \infty} s_n = \infty$ and if $(t_n)$ is a bounded sequence, then $\lim_{n\to \infty} s_n + t_n = \infty$ I know that since $(t_n)$ is bounded, $\exists$ an $M_1$ > 0, such that $(t_n)$ < $M_1$ and since $\lim_{n\to \infty} s_n = \infty$, $\forall$ n > 0, $\exists$ N($M_2$) such that n > N $\Rightarrow s_n > M_2 $.
And the definition of convergence is: $\forall \epsilon > 0, \exists$  a  number N = $N(\epsilon)$ such that $\lvert (s_n + t_n) - L\rvert < \epsilon$ where L = $\lim_{n\to \infty} (s_n + t_n) $.
We have also proven previously that if $\lim_{n\to \infty} s_n = 0$ and if $(t_n)$ is a bounded sequence, then $\lim_{n\to \infty} s_n + t_n = 0$, but I couldn't figure out how to use that to help me in this either.
I'm just not really sure how to combine all of this? The first two definitions in my notes both use M, but I'm pretty sure they aren't necessarily the same number, so I used $M_1$ and $M_2$. I'm sure I probably have to use the definiton of convergence somehow, but I'm not sure how to without having a limit to put in the equation? And assuming the $\lim = \infty$ would defeat the purpose since it's what I'm trying to prove, right?
I was able to get the inequality $M_2 + t_n$ < $s_n + t_n$ < $M_1 + s_n$, and I thought I was onto something, but I couldn't fit it into the convergence definition 
I'm always have trouble trying to figure out general proofs like this where there aren't any numbers to go off of, and I'm not sure which theorems that prove convergence I could use conversely to prove divergence?
 A: $\lim_{n\to\infty}s_n=\infty$ means for all $M>0$, there exists $N(M)$ such for $n\geq N$, we have $s_n>M$. Since $\{t_n\}$ is bounded, there exists $C\in \mathbb{R}$ such that $t_n>C$ for all $n$.
Hence, let $M>0$ be arbitrary. Since $\lim_{n\to\infty}s_n=\infty$, there exists $N$ such that for all $n\geq N$, we have $s_n>M-C$, which implies $s_n+t_n>M-C+t_n>M.$ It now follows from the definition that $\lim_{n\to\infty} s_n + t_n=\infty.$
A: You have a lot of small notational issues. For example, because $\{t_n\}$ is bounded, you know there is a $M_1$ so that $|t_n| \leq M$. Notice that you have dropped the sequence notation, it is about any element of the sequence being less than $M$ in magnitude. 
Moreover, if $\lim_{n \to \infty} s_n= 0$ and $\{t_n\}$ is bounded, it is not true that $\lim_{n \to \infty} (s_n + t_n)= 0$. For instance, take $s_n= \frac{1}{n}$ and $t_n= (-1)^n$, or $s_n= \frac{1}{n}$ and $t_n= 1+\frac{1}{n}$, etc.
As for proving this statement, think, if $s_n$ is becoming larger and larger, and the numbers $t_n$ are 'small', in that they are all less in magnitude than $M$, then the numbers $s_n+t_n$ should be becoming larger and larger in magnitude. So here is a hint, that requires just a little bit of work to be a proof. You are used to using a certain triangle inequality, but you can go the other way:
$$
|s_n+ t_n| \geq |s_n| - |t_n|
$$
Then what can you say about $|t_n|$? What can you do with the $s_n$'s? Combining this information, why can you say that $s_n + t_n$ must be becoming larger and larger?
