Suppose that Fn:[a,b]->R is a sequence of continuous functions that converges po

Question

Suppose that Fn:[a,b]->R is a sequence of continuous functions that converges pointwise to a continuous F:[a,b]-> R. Suppose that for any x in [a,b] the sequence { I Fn(x)-F(x) I } is monotone. Show that the sequence {Fn} converges uniformly. Also, this " I " means absolute value. In desperate need of some help! Please give detail. Thank you!!

Explanation / Answer

Suppose that Fn:[a,b]->R is a sequence of continuous functions that converges pointwise to a continuous F:[a,b]-> R. Suppose that for any x in [a,b] the sequence { I Fn(x)-F(x) I } is monotone. Show that the sequence {Fn} converges uniformly. Also, this " I " means absolute value. In desperate need of some help! Please give detail. Thank you!! GIVEN THAT Fn:[a,b]->R is a sequence of continuous functions that converges pointwise to a continuous F:[a,b]-> R. ...LET IT CONVERGE TO L SAY .........................................1 CONSIDER THE IN EQUALITY |[ Fn(x)-L] I 0 ....................................2.... ..WHERE E IS A SMALL POSITIVE NUMBER SPECIFIED ......... IF WE CAN SHOW THAT THE INEQUALITY 1 HOLDS FOR ALL X IN [A,B] INDEPENDENT OF THE ACTUAL VALUE OF X IN THE INTERVAL , THEN THAT WOULD MEAN THAT Fn(X) IS UNIFORMLY CONTINUOUS IN THE INTERVAL GIVEN .. FROM 1 AND THE GIVEN FACT THAT that for any x in [a,b] the sequence { I Fn(x)-F(x) I } is monotone. ...............3.... IT FOLLOWS THAT IF { I Fn(x)-F(x) I } IS INCREASING THEN IT HAS AN UPPER BOUND AND IF { I Fn(x)-F(x) I } IS DECREASING IT HAS A LOWER BOUND .. LET THE BOUND BE L .... ..THEN ....... |[ Fn(x)-L] I 0 ....................................2.... BE HELD FOR ANY VALUE OF X IN THE INTERVAL [A,B] INDEPENDENT OF ITS ACTUAL VALUE , FOR A PARTICULAR VALUE OF n= P SAY AND FOR ALL n> P WE SHALL HAVE INEQUALITY 2 SATISFIED. HENCE the sequence {Fn} converges uniformly...PROVED

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