Let ( a 1 , a 2 , . . . , a n ) ∈ R n and b ∈ R. Prove that the set: F 1 := { ( x 1 , x 2 , . . . , x n ) ∈ R n | ∑ i = 1 n a i x i ≥ b }is closed in R n

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Let   (      a    1    ,      a    2    ,  .  .  .  ,      a    n    )  ∈      R    n   and   b  ∈  R. Prove that the set:      F    1    :=  {  (      x    1    ,      x    2    ,  .  .  .  ,      x    n    )  ∈      R    n    |      ∑          i      =      1          n        a    i        x    i    ≥  b  }is closed in       R    n

Ryan Newman · Accepted Answer

First prove that if   f  (  x  )  :      R    n    →  R  is continuous then the set   {  x  ∈      R    n    |  f  (  x  )  ≥  b  } is closed in       R    n  . So let   f  (      x    1    ,      x    2    ,  .  .      x    n    )  =      ∑          i      =      1          n        a    i        x    i  . If   h  ,  k be to function which are continuous then   h  +  k is continuous. so it is enough to show that   f  (  x  )  =  a  x is continuous which is obvious (  δ  ≤      ϵ          |      a      |      )

Let ( a 1 </msub> , a 2 </msub> , . . . , a n

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