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Part BCSIR NET December 2023

Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

Consider the following subset of $R : U =$ { $x \in R : x^{2} - 9 x + 18 \leq 0, x^{2} - 7 x + 12 \leq 0$ }. Which one of the following statements is true?

A.inf U = 5.
B.inf U = 4.
C.inf U = 3.✓
D.inf U = 2.

Solution

$x^{2} - 9 x + 18 \leq 0 \Leftrightarrow x \in [3, 6]; x^{2} - 7 x + 12 \leq 0 \Leftrightarrow x \in [3, 4]$ . So U = [3, 4] and inf U = 3.

Related counterexample: aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L

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From The Real Line › Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

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