How to pick increasing numbers from the list

How to pick increasing numbers from the list.

   lst = {5, 3, 6, 2, 7, 4, 8};

out:

     {5,6,7,8}

So many interesting answers, is it possible know the index of result elements or position of elements with respect to the old "lst"?

Answer

Just to be different:

Block[{i = -∞}, Select[lst, # > i && (i = #) == i &]]

Note: Alexey Popkov points out that this solution relies on Select testing each element in turn from left to right, which is not documented.

Comments

plotting - Filling between two spheres in SphericalPlot3D

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