Hello,

I'd greatly appreciate any help with the following problem:

Let $\displaystyle l$ be a line and $\displaystyle P$ a pointnoton the line.

Prove that there is no such point $\displaystyle T_0 \in l $ such that

$\displaystyle (\forall T \in l) $_____$\displaystyle d(P,T) \leq d(P, T_0)$.

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In other words, prove that there is no point $\displaystyle T_0$ on the line, such that the distance from $\displaystyle T_0$ to $\displaystyle P$ is greater than the distances from all the other points of the line to the point $\displaystyle P$.

Just to illustrate the problem, we may take $\displaystyle T_0 \in l$, and in fact $\displaystyle d(T_1,P)\leq d(T_2,P) \leq d(T_3,P) \leq d(T_0,P)$.

However, that is not always true for $\displaystyle T_0$, for there is a point $\displaystyle T_4$ such that $\displaystyle d(T_4,P) \nleq d(T_0,P) $:

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I have tried to prove this problem by supposing that the opposite is true and then arriving at contradiction, and tried to use the triangle inequality $\displaystyle (d(A,B) \leq d(A,C) + d(C,B))$ somewhere on the way, but so far none of my attempts have been successful.

Thanks a lot!