Show that the sum of the finite limits of these two series is also finite

[rb120134]

Homework Statement

Let Lim n>infinity of an and lim n>infinity of bn be real numbers, then show that their sum
lim n>infinity (an + bn) is also a real number. Hint, you can use the triangle inequality

Relevant Equations

Triangle inequality, Lim n>infinity(an), and Lim n>infinity(bn)

In the homework I am asked to proof this, the hint says that I can use the triangle inequality.
I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,

 

[PeroK]

Homework Statement:: Let Lim n>infinity of an and lim n>infinity of bn be real numbers, then show that their sum
lim n>infinity (an + bn) is also a real number. Hint, you can use the triangle inequality
Relevant Equations:: Triangle inequality, Lim n>infinity(an), and Lim n>infinity(bn)

In the homework I am asked to proof this, the hint says that I can use the triangle inequality.
I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,

I suggest you need to take an arbitrary $\epsilon > 0$ to get started.

 

[rb120134]

I suggest you need to take an arbitrary $\epsilon > 0$ to get started.

So let's say lim n> infity an =x then for every ε>0 there exists an N such that Ian-xI<ε for every n≥N. then if lim n> infity bn=y (where x and y are real numbers, given in question) for every epsilon greater then zero we can find an N such that I bn-yI<ε for every n≥N.

we have Ian-x +bn-yI<ε where ε>0 and n being equal or greater then N.

Now with the triangle inequality Ix+yI≤ IxI + IyI that implies
Ian-x +bn-yI≤ Ian-xI + I bn-yI so if we can make

I an-x I + I bn-yI <ε we know that Ian-x +bn-yI is also smaller then epsilon.

but I don't know how to conclude the proof, that this sum is also a real number

 

[PeroK]

but I don't know how to conclude the proof, that this sum is also a real number

One approach is to figure out precisely what real number it is!

 

[PeroK]

One technical point: these are sequences, not series. A series is a sum.

 

[rb120134]

One technical point: these are sequences, not series. A series is a sum.

yes you are right

 

[WWGD]

Notice you can find terms m, m' large-enough to make each difference less than $\epsilon/2$ instead of $\epsilon$. And please try to learn Latex to make your posts more readable.

 

[Mark44]

So let's say lim n> infity an =x then for every ε>0 there exists an N such that Ian-xI<ε for every n≥N.

And please try to learn Latex to make your posts more readable.

Wait, isn't using the reply button enough!?

What @WWGD is saying about LaTeX has nothing to do with the Reply button. Here is the first line I quoted redone using LaTeX:

So let's say $\lim_{n \to \infty} a_n = x \Leftrightarrow ~\forall \epsilon > 0~ \exists N \ni ~|a_n - x| < \epsilon , \forall n \ge N$

You can click on what I wrote to see my LaTeX script for the above. There is also a LaTeX tutorial in the link at the lower left of the page.

 

[THAUROS]

What @WWGD is saying about LaTeX has nothing to do with the Reply button. Here is the first line I quoted redone using LaTeX:

So let's say $\lim_{n \to \infty} a_n = x \Leftrightarrow ~\forall \epsilon > 0~ \exists N \ni ~|a_n - x| < \epsilon , \forall n \ge N$

You can click on what I wrote to see my LaTeX script for the above. There is also a LaTeX tutorial in the link at the lower left of the page.

Thanks. My question maybe wasn't clear enough. But since it nas cost me two warnings I have now contacted the platform moderators to please check and contact the user before issuing warnings.
Just to clarify, when I first read the thread I wasn't sure why the user was being asked to learn how to use Latex. What I meant was if it wasn't enough to copy, paste, and then click on the reply button.
If my question has caused any issues, I can only say that it would have been kind to reply to me with a quick question abd I would have been happy to explain my doubt. That's all.
Thanks again.
Best regards

 

[Mark44]

But since it nas cost me two warnings I have now contacted the platform moderators to please check and contact the user before issuing warnings.

Your profile doesn't show any warnings, so I'm not sure what you're talking about, unless it is the three posts of yours that were deleted. No warnings resulted from the post deletions.

Just to clarify, when I first read the thread I wasn't sure why the user was being asked to learn how to use Latex.

The comment from WWGD, and copied by me in post #8 was directed at the OP, because what he wrote was difficult to read.

What I meant was if it wasn't enough to copy, paste, and then click on the reply button.

This completely misses the point. Inequalities such as Ian-x +bn-yI≤ Ian-xI + I bn-yI (copied directly from one of the OP's posts) are much clearer when written using LaTeX. Compare the above to this:
$$|a_n - x + b_n - y| \le |a_n - x | + |b_n - y |$$

 

[THAUROS]

I don't think I missed the point at all. I just wasn't sure how to use Latex and I asked the correct question. Other than, thanks for explaining anyways.