Html drop down box with scrollbar at options

Html drop down box with scrollbar at options

I've been trying to find a solution and example for this problem for a
while now.
We have a combobox with a lot of options, ranging from 1 up to atleast 100.
Our customers don't always use proper resolutions so for some the options
will scroll out of the screen.
What we want is to have the normal HTML combobox but when you click it
show a "box" which shows only 10 options with a scrollbar to scroll to the
other 90 options
Example:

This example is (probably) made in winforms but I want to have this as a
html/javascript(jquery). At which point the options will not be visible
unless you click the control (As a normal combobox would work)
Attempts
I have already tried putting the combobox in divs and giving it a size.
But that will make it loose the popup function and we want to keep that.
I have already tried every other post in stackoverflow that concerned
combobox's (dropdownbox) with vertical scrollbars