[SOLVED] Metric Space and Inequality

Problem:

For any two functions f and g in C([0,1],R), define

http://dammserv.loopback.nu/latex/im...731_6dd3z2.png

a. Prove that this defines a metric on C([0,1],R)

b. Prove that the following inequality relating this metric and the uniform metric:

http://dammserv.loopback.nu/latex/im...731_LkZmNt.png

c. Compare the concepts of convergence of a sequence of functions in this metric and in the uniform metric.

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Attempt:

a. First, C([0,1],R) denotes that the function http://dammserv.loopback.nu/latex/im...731_kIU2aK.png is continuous.

We need to prove that

i. http://dammserv.loopback.nu/latex/im...731_sBbGj2.png if and only if f = g. If f = g, then

http://dammserv.loopback.nu/latex/im...731_Dz2Ufs.png

ii. d*(f,g) = d*(g,f) (symmetry): This is true since http://dammserv.loopback.nu/latex/im...731_r3dYyM.png , thus http://dammserv.loopback.nu/latex/im...731_WwDTf4.png

iii. Triangle Inequality:

http://dammserv.loopback.nu/latex/im...731_1oMaAp.png

http://dammserv.loopback.nu/latex/im...731_OBNt5M.png

Thus, d*(f,g) is a metric on C([0,1],R)

b.We need to prove that

http://dammserv.loopback.nu/latex/im...731_LkZmNt.png

http://dammserv.loopback.nu/latex/im...731_0IR9EY.png

I'm unsure where to start. I try to divide both sides by (b-a) which yields

http://dammserv.loopback.nu/latex/im...731_Gh6DoX.png

Here, I use the Mean Value Theorem for integrals, since the function on [a,b] is continuous, then there exists a c such that

http://dammserv.loopback.nu/latex/im...731_kRDv4C.png

http://dammserv.loopback.nu/latex/im...731_JNfkty.png

c. I'm not sure.

Thank you for your time.