In a normed space $X$, the distance $\delta$ from a element $x\in X$ to a nonempty subset $M\subset X$ is defined to be $\delta= \inf_{\hat{y}\in M} ||x-\hat{y}||$. My lecture say: is important to know whether there is a $y\in M$ such that $\delta= ||x-y||$, for example in the Figure (a), below, no exist such $y$ . My question is Why no exist such $y$? Why this $y$ is no at the most in left? 
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juaninf
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4$M$ is an open segment, the endpoints of the segment are not in $M$. – David Mitra Jun 09 '13 at 16:31
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1@DavidMitra, honestly I'm hating your style of answering questions in comments because: 1. If answer is non trivial it is worth to be an answer not a comment. If answer is trivail then defenitely there will be people who will post an answeer. When you answer question in comments other people hesite to post an answer because it looks like plagiarism. Please follow standards of MSE and don't use comments ad hoc – Norbert Jun 09 '13 at 19:39
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I'm not at all sure what that little arrow in diagram (a) is supposed to indicate, so I'll just ignore it.
Note that if you choose any point $z\in M$, then there would be a point $y\in M$ with $y<z$ (interpreting the diagram in the obviously intended way). We would also have that the distance from $y$ to $x$ is strictly smaller than the distance from $z$ to $x$. It follows that $z\ne \hat y$. As $z$ was an arbitrary point in $M$, it follows that there is no element of $M$ that minimizes the distance to $x$.
Note that $M$ is an open line segment; $M$ does not contain the endpoints of the line segment, which is what pushes through the foregoing argument. Things here are entirely analogous to the statement "the interval $(0,1)$ has no smallest element".
David Mitra
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