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 $a_n\to -\infty$?