A question from Real and Complex Analysis (Rudin's). (1 Viewer)

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MathematicalPhysicist

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I am trying to understand theorem 1.17 in page 15-16 international edition 1987.
How do you show that [tex]\phi_n(t)[/tex] is a monotonic increasing sequence of functions?
 
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It might be easier if you note that

[tex]k_n(t)=\text{floor}(2^nt)[/tex]

and [tex]\text{floor}(2x)/2\ge \text{floor}(x)[/tex].

Then, when you want to show that [tex]\varphi_n(t)\le\varphi_{n+1}(t)[/tex] consider the cases where [tex]0\le t<n[/tex], [tex]n\le t<n+1[/tex] and [tex]n+1\le t[/tex] separately.
 

MathematicalPhysicist

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Thanks, got it, basically I only need to check for t in [0,n) the other case is trivial.
 

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