Question:
Let ($a_n$)$_{n∈\mathbb N}$ and ($b_n$)$_{n∈\mathbb N}$ be sequences such that $\lim_{n→∞}$ $(a_n)$ = ∞ and $\lim_{n→∞}$$(b_n)$ = b ∈ $\mathbb R$.
Show that
$\lim_{n→∞}$($a_n$ + $b_n$) = ∞.
My answer:
Lets assume $\lim_{n→∞}$($a_n$ + $b_n$) = p ∈ $\mathbb R$.
We are told that $\lim_{n→∞}$$b_n$ = b ∈ $\mathbb R$.
$\Rightarrow \forall \epsilon_1 $ $\exists N_1\in\mathbb R$ :n>$N_1 \Rightarrow |b_n -b| < \epsilon_1$
and $\forall \epsilon_2 $ $\exists N_2\in\mathbb R$ :n>$N_2 \Rightarrow |a_n+b_n -p| < \epsilon_2$
Set: $N=\max(N_1,N_2)$ and $\epsilon_1=\epsilon_2=\frac{\epsilon}{2}$
For n>N:
$|a_n-(p-b)|=|a_n+b_n-b_n-p+b|=|(a_n+b_n-p)-(b_n-b)|\leq |a_n+b_n-p|+|b_n-b|<\epsilon$
$\Rightarrow \lim_{n→∞} (a_n) = p-b\in\mathbb R$ which is a contradiction since $\infty \notin \mathbb R$
$\Rightarrow p\notin \mathbb R$.
Hence $\lim_{n→∞}(a_n+ b_n) = \infty$
I'm unsure if this answers the question, as I think I have only shown that $\lim_{n→∞}(a_n+ b_n) $ diverges – not specifically to $\infty$.
I feel like I need to use the definition of a sequence which diverges to infinity:
$\forall K\in\mathbb R^+$ $\exists N_K\in \mathbb N: n>N_K \Rightarrow a_n>K$
But I am unsure on how to implement this after several attempts of fiddling with the inequalities.
Best Answer
If $\lim a_n = \infty$ that means, by definition, that for any $M$ there is an $N_1$ so that if $n > N_1$ then $a_n > M$.
And $\lim b_n = b$ means, by definition that for any $\epsilon >0$ there is an $N_2$ so that if $n > N_2$ then $|b_n - b| < \epsilon$.
So if $n > \max(N_2,N_1)$ we have $a_n > M$ and $b -\epsilon < b_n <b+ \epsilon$ so $a_n + b_n > b+M - \epsilon$.
That should tell us something if $\epsilon$ is small enough.....
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For any $M$:
If we let $0 < \epsilon < 1$ and $N_2$ be the required value so $n > N_2$ means $|b_n -b| < \epsilon$.
And we can let $M' = M-b +1$ then we can find an $N_3$ be the required value value so that $n > N_3$ means $a_n >M' = M-b +1$
Then if $n > \max(N_2, N_3)$ then $b_n > b-\epsilon > b-1$ and $a_n > M' = M-b+1$ so $a_n + b_n > (M-b+1)+(b-1) = M$.
So $\lim (a_n + b_n) = \infty$.