Memory Update for an old laptop

I have an old HP Pavilion dv6 Series (model dv6-1140ej).

It has 2GB of RAM in 2 sticks. According to the specs, it supports 400Mhz DDR2 memory.

I would like to increase the amount of RAM I have.

Someone told me faster memory sticks (say 800Mhz) caot be used when only 400Mhz are supported.

This is a bummer, because for some reason 4GB of 400Mhz DDR2 costs about 3-4 times more than a mode DDR2 stick with the same size.

Is this really the case ? Can't a memory stick work in a frequency lower than its optimal ?

Here is the CrucialScan results:

Can I really use DDR3L-1600 SODIMM for this old laptop ?