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Hey guys,

I have been working on the following question:

http://imageshack.us/a/img407/4890/81345604.jpg [Broken]

For part a

f and g are continuous on I

=> there exists e > 0 and t_0 s.t.

0<|{f(t) - g(t)} - {f(t_0) - g(t_0)}| < e

using |a-b| >= |a| - |b|,

|{f(t) - g(t)} - {f(t_0) - g(t_0)}| >= |{f(t) - g(t)}| - |{f(t_0) - g(t_0)}|

and

|{f(t) - g(t)}| - |{f(t_0) - g(t_0)}| >= |{f(t) - g(t)}| - |f(t_0)| + |g(t_0)|

but i'm not sure where to go from here, I have tried a lot of different steps but I end up going in circles or getting back to the triangle inequality haha.

Does anyone have any ideas on how to do this? Is there some simple trick that I am just missing?

For part b,

Proving σ>=τ is pretty straight forward, just applying the Schwarz inequality but I'm having trouble showing that ρ>σ

How can you relate that sup to the integral expression σ?

Thanks in advance

I have been working on the following question:

http://imageshack.us/a/img407/4890/81345604.jpg [Broken]

For part a

f and g are continuous on I

=> there exists e > 0 and t_0 s.t.

0<|{f(t) - g(t)} - {f(t_0) - g(t_0)}| < e

using |a-b| >= |a| - |b|,

|{f(t) - g(t)} - {f(t_0) - g(t_0)}| >= |{f(t) - g(t)}| - |{f(t_0) - g(t_0)}|

and

|{f(t) - g(t)}| - |{f(t_0) - g(t_0)}| >= |{f(t) - g(t)}| - |f(t_0)| + |g(t_0)|

but i'm not sure where to go from here, I have tried a lot of different steps but I end up going in circles or getting back to the triangle inequality haha.

Does anyone have any ideas on how to do this? Is there some simple trick that I am just missing?

For part b,

Proving σ>=τ is pretty straight forward, just applying the Schwarz inequality but I'm having trouble showing that ρ>σ

How can you relate that sup to the integral expression σ?

Thanks in advance

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