Triangle inequality

Definition

The triangle inequality states that for any real numbers $a$ and $b$, the absolute value of their sum is less than or equal to the sum of their absolute values: $|a + b| \leq |a| + |b|$. This principle is fundamental in analysis, particularly in proving properties of limits.

5 Must Know Facts For Your Next Test

1. The triangle inequality can be used to show the boundedness of sequences and functions.
2. It is essential for proving results related to the convergence of sequences and series.
3. A common application is in establishing $\epsilon-\delta$ proofs for limits.
4. The reverse triangle inequality, $||a| - |b|| \leq |a - b|$, is also frequently used.
5. Understanding the triangle inequality helps in grasping the concept of metric spaces.

Review Questions

• How does the triangle inequality assist in proving that a sequence converges?
• Can you apply the triangle inequality to show that if $\lim_{n \to \infty} a_n = L$ and $\lim_{n \to \infty} b_n = M$, then $\lim_{n \to \infty} (a_n + b_n) = L + M$?
• Explain how the reverse triangle inequality can be derived from the standard triangle inequality.

Related terms

Absolute Value: A measure of magnitude without regard to direction; for a real number $x$, denoted as $|x|$.

$\epsilon-\delta$ Definition: A formal definition of a limit; it states that a function $f(x)$ approaches a limit $L$ as $x$ approaches a point if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$.

Convergence: The property of a sequence or series approaching some limit as its index or summation goes to infinity.