First of all if you read this and the latex is all messed upp I am probably working on getting it right so plz be patient till I get it right. No need to post a comment that it doesnt work. Thanks(adsbygoogle = window.adsbygoogle || []).push({});

I havent taken a pure maths class in over 2,5 years so I can hardly remember how to write proofs:yuck:

Problem 1.

Let [tex] C \in \mathbb{R} [/tex] be a arbitrary number. Show that the function

[tex] f:[a,b]\rightarrow \mathbb{R} [/tex]

given by [tex] f(t)=cos(ct) [/tex]

is of bounded variation. i.e it satisifies the condition

[tex]Sup V_f (t) < \infty [/tex]

Proof.

[tex] V_f (t) = \lim_{n\rightarrow \infty} Sup_{ t_k^n,t_{k-1}^n \in \pi_n }

\sum_{k=1}^n |{(f(t_k^n)-f(t_{k-1}^n)}| [/tex]

with [tex] \pi_n : t_0^n < ...... < t_n^n [/tex]

Since [tex] Cos(ct) [/tex]

is differentiable we can rewrite [tex]V_f (t)[/tex] with the mean value theorem

[tex] V_f (t) = \lim_{n\rightarrow \infty} Sup_{ t_k^n,t_{k-1}^n \in \pi_n }

\sum_{k=1}^n \|{(f(t_k^n)-f(t_{k-1}^n)}| = \lim_{n\rightarrow \infty} Sup_{ t_k^n,t_{k-1}^n \in \pi_n } \sum_{k=1}^n |f^{'} (G)| (t_k^n - t_{k-1}^n) [/tex]

wich is equal to(according to the definition of the riemann integral)

[tex] \int_{a}^{b} |f^{'} (x)| dx [/tex]

with [tex]f(x)=f(t)=cos(ct) [/tex] we get

[tex] \int_{a}^{b} |-csin(ct)| dx \leq \int_{a}^{b} |c| dx = |c|(b-a)[/tex]

So

[tex] Sup V_f (t) = Sup |c|(b-a) <\infty [/tex]

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# Homework Help: Are these proofs correct(bounded and finite variation).

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