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

More Understanding sup and inf

For A ¹ Æ, it is useful to note the working characterizations of supA and infA:

For A ¹ Æ, supA is the number a such that ì ï í ï î For every x Î A, x £ a, For every e > 0, there is an x Î A such that x > a- e.

For A ¹ Æ, infA is the number b such that ì ï í ï î For every x Î A, x ³ b, For every e > 0, there is an x Î A such that x < b+ e.

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

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

·

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

inf f + inf g £ inf (f + g).

·

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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