Yes a filter is required in order to cycle, it's where most of your beneficial bacteria will colonize and live. If you're getting the Tetra Whisper 10i (I'm assuming) it'll work just fine, might be a bit strong for your Betta but you can always baffle it.
Anyways, you're asking how to cycle? Cycling is very easy all you have to do is make sure you have the following:
- Find an ammonia source, this can be from pure ammonia, fish food, fish themselves, etc.
- Get a test kit which contains: Ammonia, Nitrite, and Nitrate. These are very important as they will tell you when to do a water change and when you're fully cycled. Bottled test products are recommended as test strips are inaccurate.
Anyways, after you have those two it's time to decide to choose from either fish-in or fish-less cycle, I recommend fish-less cycle so that way no can be harmed but if you already have your Betta in the 5g and don't have anywhere else for him, fish-in cycle it is! Bettas are hardy fish so they can usually tolerate somewhat.
Anyways, I'll assume you're doing a fish-in cycle. First off whenever ammonia or nitrIte hits 0.25ppm do a water change. Keep testing ammonia, nitrIte everyday and check nitrAte every three days or so. When you're truly cycled you should have no ammonia, no nitrItes and around 10 ~ 20ppm of nitrAtes.
That's pretty much it, here's
a more in-depth guide is I wasn't clear enough.
Also to add that duckweed does not need any care at all, just leave it there. It's a weed and it'll overrun your tank in no time, so just throw some out when you see too much. :]