# Help proving that $\lim(s_n+t_n)=+\infty$

If $$\lim s_n=+\infty$$ and if $$(t_n)$$ is a bounded sequence, then $$\lim(s_n+t_n)=+\infty$$.

My proof:

Suppose $$\lim s_n=+\infty$$ and $$(t_n)$$ is bounded.

Since $$\lim s_n=+\infty$$, by definition we have that for all $$M\in\mathbb{R}$$, there exists $$N\in\mathbb{N}$$ such that $$n\geq N$$ implies that $$s_n>M$$.

Since $$(t_n)$$ is bounded, we have that for all $$n\in\mathbb{N}$$, $$k\leq t_n\leq k'$$ for some $$k,k'\in\mathbb{R}$$.

This is where I get stuck. I'm not sure how to put these two definitions together coherently. It's obvious to me that $$t_n$$ being bounded means that $$s_n + t_n >M$$. However, I'm having trouble arriving at this result from the definitions that I've stated.

Maybe since $$\lim s_n=+\infty$$, I could say that $$s_n>M-t_n$$ which implies that $$s_n+t_n>M$$. But I'm not sure how the definition of $$t_n$$ being bounded plays into this.

Am I on the right track? Any hints/advice would be helpful. Thank you!

• I think you gave the definition of $s_n$ correctly. Maybe since $s_n$ is unbounded, we have that at some point $P$, $s_n>t_n$. Then concluding that $s_n+t_n$>$M$? Feb 27 at 23:10
• Try working with a constant with regards to the bounded sequence. More specifically, show that $\lim_{n \rightarrow \infty} [s_n + k]$ diverges. Feb 27 at 23:30

You want to prove that $$\lim (s_n + t_n) = \infty$$.

Given some $$A \in \mathbb{R}$$.

We know that there exists $$K$$ such that $$|t_n| < K$$ for all $$n$$.

Since $$\lim s_n = \infty$$, we can pick $$N$$ such that $$n\ge N\implies s_n \ge A+K$$.

Thus, $$n\ge N \implies s_n+t_n\ge A \ \ \blacksquare$$.

Comment: I used to struggle with these proofs as well. Make sure to think about what exactly you want to prove. I always start these proofs with "given some $$A$$ / given some $$\epsilon$$ (as appropriate)". Consider the definition of the statement as a "challenge": if an adversary gives you some $$A$$, how do you come up with $$N$$ that beats the challenge?

The way you did it almost worked (you got $$s_n>M-t_n$$). You just have to see that $$|t_n|.

• Ok, cool I think this is exactly what I was looking for. An equivalent way of saying that $t_n$ is bounded is that $|t_n|<K$? Feb 27 at 23:22
• Well, you should say "there exists K such that...". Feb 27 at 23:23
• haha ya I know. I guess I was just blanking on the fact that since its bounded, obviously there exists a $K$ such that $-K<t_n<K$, which is to say that $|t_n|<K$. Thanks for the help! :) Feb 27 at 23:24
• "$(t_n)$ is bounded" is equivalent to, "There exists K such that $|t_n|<K$ for all $n$". And +1 for this answer: I would have written the proof in the exact same way. Feb 27 at 23:25
• In terms of the variables in the original question, it's sufficient to choose $N$ such that whenever $n \ge N$, then $s_n \ge A - k$ (with the thought in the back of one's mind that $k$ might be a "very negative" number, but the boundedness of $t_n$ still ensures that $k$ remains "finitely negative"). Feb 28 at 1:14

We can do a proof by “ contradiction “. Assume $$s_n + t_n \to L < \infty$$, then $$s_n + t_n$$ is bounded, say by $$M$$. Thus $$|s_n| = |s_n + t_n - t_n| \le |s_n + t_n| + |t_n| \le M + T$$. So $$s_n$$ is bounded, but it is unbounded since $$s_n \to \infty$$. Therefore $$s_n + t_n \to \infty$$.

• And what about the case where $s_n + t_n$ does not approach infinity, but not because it has a finite limit? (For example, to rule out the possibility that $s_n + t_n = (-1)^n n$. Or the possibility that $s_n + t_n = (-1)^n$.) Feb 28 at 1:16
• @Daniel Schepler: I thought about the case you brought up but assumed OP’s sequence is either convergent to $\infty$ or to $L < \infty$. Feb 28 at 2:23

You got a definition of a limit to infinity, check for $$s_n+t_n$$, that is

$$\forall _{M \in \mathbb{R}} \exists _{N \in \mathbb {N}} \forall _{n \geq N} \; s_n + t_n > M \;$$ Because...

$$\forall _{M \in \mathbb{R}} \exists _{N \in \mathbb {N}} \forall _{n \geq N} \; s_n > M \hspace{30px} \exists _{k,k' \in \mathbb{R}} \forall _{n \in \mathbb {N}} \; k \geq t_n \geq k' \Rightarrow \exists _{K \in \mathbb{R}} \forall _{n \in \mathbb{N}} \; |t_n| < K$$

$$\Rightarrow \forall _{M \in \mathbb{R}} \exists _{N \in \mathbb{N}}\forall _{n \geq \mathbb{N}} \; s_n > M + K \Rightarrow s_n + t_n > s_n - K > M \quad$$ Since $$\quad -K < t_n < K$$