- #1
math2006
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Let [tex]f[/tex] be a real uniformly continuous function on the bounded
set [tex]A[/tex] in [tex]\mathbb{R}^1[/tex]. Prove that [tex]f[/tex] is bounded
on [tex]A[/tex].
Since f is uniformly continuous, take [tex]\epsilon = m, \exists \delta > 0[/tex]
such that
[tex]|f(x)-f(p)| < \epsilon [/tex]
whenever [tex]|x-p|<\delta[/tex] and [tex]x,p \in A[/tex]
Now we have
[tex]|f(x)| < m + |f(p)| [/tex]
Obviously i should show [tex]b + |f(p)|[/tex] is bounded, but no idea how.
Could someone help me? thx
set [tex]A[/tex] in [tex]\mathbb{R}^1[/tex]. Prove that [tex]f[/tex] is bounded
on [tex]A[/tex].
Since f is uniformly continuous, take [tex]\epsilon = m, \exists \delta > 0[/tex]
such that
[tex]|f(x)-f(p)| < \epsilon [/tex]
whenever [tex]|x-p|<\delta[/tex] and [tex]x,p \in A[/tex]
Now we have
[tex]|f(x)| < m + |f(p)| [/tex]
Obviously i should show [tex]b + |f(p)|[/tex] is bounded, but no idea how.
Could someone help me? thx
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