MthT 430 Projects Chap 8e - More Adding sup and inf

More Understanding sup and inf

For A ≠ ∅, it is useful to note the working characterizations of supA and infA:

For A ≠ ∅, supA is the number α such that ⎧ ⎪ ⎨ ⎪ ⎩ For every x ∈ A, x ≤ α, For every ϵ > 0, there is an x ∈ A such that x > α− ϵ.

For A ≠ ∅, infA is the number β such that ⎧ ⎪ ⎨ ⎪ ⎩ For every x ∈ A, x ≥ β, For every ϵ > 0, there is an x ∈ A such that x < β+ ϵ.

If f is a bounded function on [0,1], we define

sup f = sup x ∈ [0,1]  f(x) inf f = inf x ∈ [0,1]  f(x)

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Show that if f and g are bounded functions on [0,1], then

sup (f + g) ≤ sup f + sup g.

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Give an example of bounded functions f and g on [0,1] such that

sup (f + g) < sup f + sup g.

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Show that if f and g are bounded functions on, then [0,1],

inf f + inf g ≤ inf (f + g).

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Show that if f and g are bounded functions on, then [0,1],

inf f + sup g ≤ sup (f + g).

The general result is that for two bounded bounded functions f and g on [0,1]

inf f + inf g ≤ inf (f + g) ≤ inf f + sup g ≤ sup (f + g) ≤ sup f + sup g.

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