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I am trying with no luck to prove:

Let (X,d) be a metric space and A a non-empty subset of X. For x,y in X, prove that

d(x,A) < d(x,y) + d(y,A)

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    You won't be able to prove this with strict inequality; for instance, take $A = \{y\}$. You need $\leq$.2012-02-12
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    Try drawing a picture of a set A with two points x and y outside it.2012-02-12
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    It will help if you have sitting in front of you the definition of distance from a point to a set.2012-02-12
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    I've drawn all the pictures. But doesn't it come down to some creative trick or using a fact about infimum? I can't see it.2012-02-12

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