Section 21: Problem 5 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Theorem. Let $x_{n}\rightarrow x$ and $y_{n}\rightarrow y$ in the space $\mathbb{R}$ . Then $\begin{eqnarray*} x_{n}+y_{n} & \rightarrow & x+y,\\ x_{n}-y_{n} & \rightarrow & x-y,\\ x_{n}y_{n} & \rightarrow & xy, \end{eqnarray*}$ and provided that each $y_{n}\neq0$ and $y\neq0$ , $x_{n}/y_{n}\rightarrow x/y.$ [Hint: Apply Lemma 21.4; recall from the exercises of §19 that if $x_{n}\rightarrow x$ and $y_{n}\rightarrow y$ , then $x_{n}\times y_{n}\rightarrow x\times y$ .]

Using the hint, Exercise 6 of §19 shows that $x_{n}\times y_{n}\rightarrow x\times y$ , and we may now apply Lemma 21.4 and Theorem 21.3.

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Section 21: Problem 5 Solution

Section 21: Problem 5 Solution