Proving Equivalence of f(x) and (1/n) Summation of f(x_k)

[beebeeamoras]

Q1. f is a continuous real valued function on [o,oo) and a is a real number
Prove that the following statement are equivalent;
(i) f(x)--->a, as x--->oo
(ii) for every sequence {x_n} of positive numbers such that x_n --->oo one has that
(1/n)\sum f(x_k)--->a, as n--->oo (the sum is taken from k=1 to k=n)

 

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