Let a_1, a_2, ..., a_n be real numbers with |a_k| lessthanorequalto M for each k

ID: 3108892 • Letter: L

Question

Let a_1, a_2, ..., a_n be real numbers with |a_k| lessthanorequalto M for each k (M is a constant). Show that if |a_1 + a_2 + ... + a_n/n| = M, then a_1 = a_2 = ... = a_n. Show the same thing as in part (a), but this time assume that a_1, a_2, ..., a_n are complex numbers. This is the "continuous version" of part (a). Let f: [a, b] rightarrow R be a continuous function with |f(x)| lessthanorequalto M for each x Element [a, b]. Show that if |1/b - a integral^b_a f(x) dx| = M, then f must be constant.

Explanation / Answer

Given,

|(a1+a2+.....+an)/n|=M

therefore,

|a1+a2+....+an|=M×n

Given,for each 'k' ,M is constant

I.e. |ak|<=M

n= number of terms

a1+a2+....+an=M×n

As n= number of terms and M is constant for all a1,a2,a3,......,an.

There must be,

a1=a2=a3=…………=an.

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Let a_1, a_2, ..., a_n be real numbers with |a_k| lessthanorequalto M for each k

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