Why do the limits of the left endpoints $a$ and right endpoints $b$ coincide if the length of the nested intervals approaches zero?

Conditions

• $\{a_n\}$ and $\{b_n\}$ are convergent sequences with limits $a$ and $b$.
• The length of the intervals $b_n - a_n$ approaches 0 as $n \to \infty$.

Reasoning, step by step

1. Establish that $\lim a_n = a$ and $\lim b_n = b$ exist due to monotonicity and boundedness.
2. Express the interval length as the difference between endpoints: $d_n = b_n - a_n$.
3. Apply the limit laws for differences of convergent sequences: $\lim d_n = \lim b_n - \lim a_n = b - a$.
4. Use the given condition that $\lim d_n = 0$.
5. Solve $b - a = 0$ to find $a = b = \xi$.

Example

Consider intervals $[0, 1/n]$. Left limit $a=0$, Right limit $b=0$. Length $1/n \to 0$. Here $a=b=0$.

Common misconceptions

• Assuming $a$ and $b$ are always equal regardless of interval length; they are only equal if the gap closes to zero.
• Confusing the limit of the difference with the difference of the limits (though valid here due to convergence).

Watch the explanation

Explore next

• What happens if the interval lengths approach a positive constant $c > 0$?
• How does this relate to the completeness of the real number system?