Problem with with the sum of a divergent sequence $\rightarrow +\infty$ and a lower bounded sequence Im trying to prove that if $a_n\rightarrow+\infty$ and $b_n$ is lower bounded only, then $a_n + b_n\rightarrow+\infty.$ I have to use the definition of a divergent sequence but I'm stuck. $\forall M > 0,\ \exists ν\in N : an > M\forall n > ν.$ That is the definition that I'm using. I know that I have to use the fact that $b_n$ is a lower bounded sequence so $∃A<b_n.$ Can some help?
 A: Here my answer if i have well undestood your question. But take into account that i am just a student.
First let recall that by definition: $\lim_{n\rightarrow \infty }a_n= \infty$ means that: $\forall M \in \mathbb{R} , \exists N \in \mathbb{N} \; s.t. \; \forall n > N \Rightarrow a_n> M$ (i)
On the other hand it is given that $b_n$ is lower bounded. That means $\exists A \in \mathbb{R} \; s.t. \; \forall n \in \mathbb{N} \Rightarrow b_n > A.$ (ii)
Now if we writte $c_n = a_n + b_n$ it cames that for all $M' \in \mathbb{R} $ we can define $M'-A$ for which (according to (i) ) , it exists at least one $N' \in \mathbb{N} $ s.t. $a_n > M' - A$.
So we have $\forall n \in \mathbb{N}> N' \Rightarrow c_n=a_n+b_n>M'-A + b_n$.
But by (ii) we know too that $\forall n \in \mathbb{N} \Rightarrow b_n>A$, hence if it is true $\forall n \in \mathbb{N}$ it is in particular true $\forall n \in \mathbb{N}> N'$.
To conclude we have $\forall M' \in \mathbb{R} , \exists N'$,as define from $a_n$ just above, s.t. $\forall n > N' \Rightarrow c_n > M'$.
And this is the definition of: $\lim_{n\rightarrow \infty }c_n=\lim_{n\rightarrow \infty }a_n + b_n = \infty$
Q.E.D.
