Exercise 4.12

Question

Exercise 12:
    Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $Y$, and suppose $g$ is a uniformly continuous mapping of $Y$ into a metric
    space $Z$. Then show that $g\circ f$ is a uniformly continuous mapping of $X$ into $Z$.

analambanomenos · Accepted Answer

\def\eps{\varepsilon} Let $\varepsilon>0$. Then there is $\eta>0$ such that $d_Z\bigl(g(x),g(y)\bigr)<\varepsilon$ if $d_Y(x,y)<\eta$. There is also $\delta>0$ such that $d_Y\bigl(f(x),f(y)\bigr)<\eta$ if $d_X(p,q)<\delta$. Hence if $d_X(p,q)<\delta$, then $d_Z\bigl(g(f(p)),g(f(q))\bigr)<\varepsilon$, so $g\circ f$ is uniformly continuous.

Exercise 4.12

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Exercise 4.12

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