Question

Show that the following sets are closed in R²: (1) A = {(x, y) ? R² : xy = 1}; (2) B = {(x, y) ? R² : x² + y² ? 1}.

          Show that the following sets are closed in R²:
(1) A = {(x, y) ? R² : xy = 1};
(2) B = {(x, y) ? R² : x² + y² ? 1}.

Added by Julie S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Narayan Hari

Step 1

Let ${(x_n, y_n)}$ be a sequence in A such that $(x_n, y_n) \to (x, y)$ as $n \to \infty$. This means that $x_n \to x$ and $y_n \to y$ as $n \to \infty$. Since $(x_n, y_n) \in A$, we have $x_n y_n = 1$ for all $n$. Now, we want to show that $(x, y) \in A$. Show more…

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