Triangle inequality with countably infinite terms

In summary, the lecture in the real analysis course discussed the proof that absolute convergence of a series implies convergence. The incorrect proof was shown first, using the triangle inequality, but the correct proof was then demonstrated using the Cauchy convergence of the sequence of partial sums. However, the question was raised about why the first proof is wrong, as the triangle inequality is true for all natural numbers. A counterexample was given to illustrate that the logic used in the first proof is not necessarily correct.
  • #1
terhorst
11
0
In lecture in my real analysis course the other day we were proving that absolute convergence of a series implies convergence. Our professor started off by showing us the wrong way to prove it:

[tex]\left| \sum_{k=1}^\infty a_k \right| \leq \sum_{k=1}^\infty \left| a_k \right| < \epsilon[/tex]

Then he demonstrated the correct proof, by showing that the sequence of partial sums is Cauchy convergent and then using the triangle inequality.

But this got me thinking: why is the first proof wrong? I definitely agree that the second proof is more solid, but if the triangle inequality is proved by induction, meaning it's true for all natural numbers, isn't that, well, infinite? I was wondering if someone could supply a counterargument or proof by contradiction illustrating why this conclusion is incorrect. Thanks as always.
 
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  • #2
It's true for any natural number. Infinity isn't a natural number. That about sums it up.

Obviously the conclusion that the triangle inequality holds for an infinite series holds, since you proved it did. But that doesn't mean the logic that you used to make that conclusion is necessarily correct.

Here's an example that might elucidate the problem:

Every sequence has an element of greatest magnitude. This is obviously true for finite sequences [tex]a_1, a_2,..., a_n[/tex] and is [tex]max_{i=1,...n}(| a_i |)[/tex]

Hence, by your logic, if we have an infinite sequence [tex]a_1, a_2,... [/tex] then [tex]max_{i=1,2,...}( |a_i| )[/tex] exists and is in the sequence. Obvious counterexample: [tex]a_i = 1-\frac{1}{i}[/tex]
 

1. What is the definition of Triangle inequality with countably infinite terms?

Triangle inequality with countably infinite terms is a mathematical concept that states that the sum of any two sides of a triangle must be greater than the length of the third side. This concept can also be applied to an infinite sequence of numbers, where the sum of any two consecutive terms must be greater than the next term.

2. How is Triangle inequality with countably infinite terms different from the traditional Triangle inequality?

The traditional Triangle inequality only applies to three sides of a triangle, while Triangle inequality with countably infinite terms can be applied to an infinite sequence of numbers. Additionally, Triangle inequality with countably infinite terms is a more general concept and encompasses the traditional Triangle inequality as a special case.

3. What is the significance of Triangle inequality with countably infinite terms in mathematics?

Triangle inequality with countably infinite terms is an important concept in mathematics as it helps to prove the convergence of infinite sequences and series. It is also used in various mathematical proofs and has applications in fields such as analysis, number theory, and geometry.

4. Can Triangle inequality with countably infinite terms be applied to any sequence of numbers?

No, Triangle inequality with countably infinite terms can only be applied to sequences of numbers that satisfy certain conditions. For example, the sequence must be bounded and monotonically increasing or decreasing for the inequality to hold.

5. How is Triangle inequality with countably infinite terms used in real-world applications?

Triangle inequality with countably infinite terms has various real-world applications, such as in signal processing, where it is used to analyze signals and determine their properties. It is also used in the study of fractals and in computer science algorithms, such as the sorting algorithm.

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