Why must a locally compact second countable Hausdorff space be second countable to imply paracompactness? The textbook version of the result I've seen states: A locally compact second countable Hausdorff space is paracompact. Is the property of being second countable needed, or have I missed something? 
My thinking:


*

*If the space is locally compact then each point has a compact neighborhood.

*For this compact neighborhood each covering has a finite sub-covering.

*The finite sub-covering is a locally finite refinement. 
Thanks in advance.

Without the second countable condition the result is not true. The long line is locally compact Hausdorff and is not paracompact because it is locally metrizable but not metrizable.
