Let f 0 infinity rightarrow R be a continuous function with

Let f: [0, infinity) rightarrow R be a continuous function with f(0) = 2 and lim_n rightarrow infinity f(n) = 0. Prove that there exists a c (0, infinity) such that f (c) = 1.

Solution

Consider the sequence:a_n=f(n)

a_n converges to 0

Hence there exist N so that

|a_n|<1 for all n>N

ie f(n)<1 for all n>N

But f is a continuous function

f(0)=2

f(N+1)=d<1

Hence by Intermediate value theorem there exist a c in the interval (0,N+1)

so that f(c)=1

Hence proved.