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

[MathematicalPhysicist]

I am trying to understand theorem 1.17 in page 15-16 international edition 1987.
How do you show that $$\phi_n(t)$$ is a monotonic increasing sequence of functions?

 

[yyat]

It might be easier if you note that

$$k_n(t)=\text{floor}(2^nt)$$

and $$\text{floor}(2x)/2\ge \text{floor}(x)$$.

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

 

[MathematicalPhysicist]

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