# Infinite limit of the sum of two sequences (one diverges to infinite and the other converges)

We have $$\{a_n\}_{n\in N}$$ $$\{b_n\}_{n\in N}$$ both sequences of real numbers.

I have to prove that:

$$\lim_{n\to\infty} a_n = l \in R , \lim_{n\to\infty} b_n = +\infty \Longrightarrow\lim_{n\to \infty}(a_n +b_n) = +\infty$$

Everything with the limit's definition,

$$\lim_{n\to\infty} a_n = l \Longleftrightarrow \forall\epsilon > 0 , \exists n_o\in N , \forall n>n_0 : |a_n-l|<\epsilon$$

$$\lim_{n\to\infty} b_n = +\infty \Longleftrightarrow \forall M\in R, \exists n_0\in N,\forall n>n_0 : b_n>M$$

What can I do?

I tried writing $$|a_n-l|<\epsilon$$ as $$-\epsilon to get that $$a_n <\epsilon + l$$ and $$a_n> -\epsilon + l$$

Set $$M = M+\epsilon - l$$

Then: $$b_n >M+\epsilon - l$$ and $$a_n> -\epsilon + l$$

Every item on the previous equations are real numbers, which are partially ordered, so we can add those two equations and get:

$$b_n +a_n>M+\epsilon - l -\epsilon +l$$

Which equals to $$b_n +a_n>M$$

In this case, we want to prove that $$\forall M>0:\exists n_0\in\mathbb{N}:\forall n \geqslant n_0: a_n+b_n > M$$. In terms of proving it, we'll want to choose an $$M$$, after which we construct an $$n_0$$ with what we're given, and most likely depending on the $$M$$ we've chosen. Then we show that that $$n_0$$ gives us what we want. In your question you seem to forget about the $$n_0$$, which is probably where you're stuck.
Proof: Choose a real $$M$$, then we can see that there exists an $$n_{0,a} \in \mathbb{N}$$ such that for all $$n\geqslant n_{0,a}$$, $$|a_n-l|<1$$ (The 1 can be an arbitrarily chosen $$\epsilon >0$$). From the last inequality follows that for such $$n$$, $$a_n>-1+l$$.
Likewise, there exists an $$n_{0,b}\in \mathbb{N}$$ such that for all $$n\geqslant n_{0,b}$$, $$b_n>M+1-l$$.
Now, let $$n_0 = \mathrm{max}(n_{0,a},n_{0,b})$$, then for $$n\geqslant n_0$$, we get both $$a_n>-1+l$$ and $$b_n>M+1-l$$. We can add these equalities to get $$a_0+b_0>M$$.