- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- Prove that $\lim_{t\to p} F(t) = A$ if and only if $\lim_{t\to p} \|F(t) - A\| = 0.$
- If $\lim_{t \to\ p}\|F(t) - A\| = 0,$ then it must be the case that $\lim_{t \to\ p}\|F(t) - A\|^2 = 0$
- If $\lim_{t \to\ p}F(t) = A,$ then it follows that $\lim_{t \to\ p}\|F(t) - A\| = 0.$ Now, suppose that $\lim_{t \to\ p}\|F(t) - A\| = 0,$ this means that $\lim_{t \to\ p}\|F(t) - A\|^2 = 0,$ or in other words: $$\lim_{t \to\ p}\left[F(t) - A\right] \cdot \left[F(t) - A\right] = 0$$But this implies that $C = \left[F(t) - A\right] \to 0$ as $t \to p.$ In other words, $$\lim_{t \to \ p} F(t) = A$$Thus, we have shown that $\lim_{t\to p} F(t) = A$ if and only if $\lim_{t\to p} \|F(t) - A\| = 0. \quad \blacksquare$