Logical trap in R topology We know that in metric spaces, Bolzano-Weierstrass (BW) (each infinite set owns a cluster point) and Borel-Lebesgue (BL) properties are equivalent, i.e. compactness and countably compactness are equivalent (and also sequential compactness).
In the set of real numbers $\mathbb{R}$, the BW theorem is often written : "Any bounded infinite set has a cluster point". So according to the equivalence, we can say that in $\mathbb{R}$, "any bouded set verifies (BL)", which is not true because we know that only bounded and closed sets do...
Can you help me to find where I am wrong among this...?
EDIT : to be more precise, here is the wrong reasoning: 
"In $\mathbb{R}$, the collection of all infinite bounded sets verifies the (BW) property; since this property is equivalent to the (BL) property and since this one means compactness, all infinite bounded sets is compact."

A sequence of points in a bounded but not closed subset will have a cluster point in $\mathbb R$, but the cluster point may fail to be a member of the subset -- so it is not necessarily sequentially compact.
