Suppose $a_n \rightarrow +-\infty$ and $(b_n)$ is bounded. Show that $a_n+b_n \rightarrow +-\infty$. I tried this:
$|a_n|\rightarrow +-\infty$, so $|a_n+-\infty|<\epsilon$. It is also true that $|b_n|
Regards, Kevin
Suppose $a_n \rightarrow +-\infty$ and $(b_n)$ is bounded. Show that $a_n+b_n \rightarrow +-\infty$. I tried this:
$|a_n|\rightarrow +-\infty$, so $|a_n+-\infty|<\epsilon$. It is also true that $|b_n|
Regards, Kevin
I wouldn't advise you to add/subtract infinity until you'll have enough experience in this.
The strict proof is like this:
suppose that $a_n\to+\infty$, so for any $E>0$ (here we are especially interested in large values of $E$) there exists $N(E)$ such that $a_n>E$ for all $n\geq N$.
As you have written, there is a constant $M$ such that $|b_n|
To prove that $a_n+b_n\to+\infty$ we should show that for any E' there is N(E') such that for all n\geq N(E') holds a_n+b_n>E'.
We can clearly do it: pick up any E', then $a_n+b_n>a_n-M$ (see 2.), hence to make a_n+b_n>E' we just need to make a_n>E'+M for any E' - and that will be sufficient (do you agree here?)
Based on 1., we just take N(E'+M) so a_n>E'+M for all n\geq N(E'+M), hence a_n+b_n>E' for all n\geq N(E'+M) and hence $a_n+b_n\to +\infty$.
Could you please follow the same steps to prove the case when $a_n\to -\infty$?