let I =[a,b] a<b.

Since f is uniformly continuous on [a,b] we have :

for all ε>0 there exists a δ>0 and such that:

for all x.yε[a,b] and |x-y|<δ ,then |f(x)-f(y)|<ε........................................... ........................................1

ΝΟW subdivide [a,b] into,n subinterval of equal lengths less than or equal to δ,as follows:

Choose .This is possible since b-a and δ are fixed positive Nos.

Choose points such that

,i = 1,2,3.............n [ the inequality follows from the choice of

Then (1) implies that:

and hence

Similarly

.................................................. ................................2

Repeating this argument for each successive subinterval up to the nth subinterval, we obtain:

.

Hence ,if we choose we obtain:

,

hence f is bounded on I