Let f : [ a , b ] → R be a continuous function such that for each x ∈ [ a , b ] there exists y ∈ [ a , b ] such that | f ( y ) | ≤ 1 2 | f ( x ) | . Prove that there exists c ∈ [ a , b ] such that f ( c ) = 0.I am trying to use intermediate value theorem for continuous function. Here | f ( x ) | is maximum with respect to | f ( y ) | . Hence | f ( y ) | ≤ 1 / 2 ( | f ( x ) | + | f ( y ) | ) ≤ | f ( x ) | .Now I can say f ( c ) = 1 / 2 ( | f ( x ) | + | f ( y ) | for some c ∈ ( a , b ) . But how to show f ( c ) = 0 ?

Question

Let   f  :  [  a  ,  b  ]  →      R   be a continuous function such that for each   x  ∈  [  a  ,  b  ] there exists   y  ∈  [  a  ,  b  ] such that       |    f  (  y  )      |    ≤      1    2        |    f  (  x  )      |    . Prove that there exists   c  ∈  [  a  ,  b  ] such that   f  (  c  )  =  0.I am trying to use intermediate value theorem for continuous function. Here       |    f  (  x  )      |   is maximum with respect to       |    f  (  y  )      |    . Hence      |    f  (  y  )      |    ≤  1      /    2  (      |    f  (  x  )      |    +      |    f  (  y  )      |    )  ≤      |    f  (  x  )      |    .Now I can say   f  (  c  )  =  1      /    2  (      |    f  (  x  )      |    +      |    f  (  y  )      |   for some   c  ∈  (  a  ,  b  )  . But how to show   f  (  c  )  =  0  ?

Jayvion Mclaughlin · Accepted Answer

Let       x    0    ∈  [  a  ,  b  ]. Then there is       x    1    ∈  [  a  ,  b  ] such that       |    f  (      x    1    )      |    ≤      1    2        |    f  (      x    0    )      |    .Now we get inductively a sequence   (      x    n    ) in   [  a  ,  b  ] with(*)       |    f  (      x    n    )      |    ≤      1          2      n            |    f  (      x    0    )      |    .  (      x    n    ) contains a convergent subsequence   (      x                  n        k              ) with limit   c  ∈  [  a  ,  b  ].  f is continuous, thus   f  (      x                  n        k              )  →  f  (  c  ) .From (*) we get:   f  (      x                  n        k              )  →  0.Consequence:   f  (  c  )  =  0