Just a distinction - for analog cams it is a DVR and it is an NVR for ip cameras - so make sure that $50 unit is indeed an NVR.
P2P is simply a method to make a camera "plug and play" for the consumer where they can simply scan a code and it pulls up on an app, but it needs an internet connection initially (P2P should be avoided BTW for security concerns), but many cameras have that and you can still use it without an internet connection. Most the cameras people use on this site have P2P capability and we have turned that off as our cams are not on the internet. In all honesty, with very little computer knowledge, most can set these up without P2P. Or you configure the system initially connected to the internet, let P2P connect it, and then disconnect and hang up the cameras and place the NVR where you want to (but try to avoid that if possible).
Look for an IP cam that has ONVIF capability and in all likelihood it means it can work offline like you intend. Anything else is proprietary and will need an internet connection like Ring, Nest, Arlo, etc.
I found this very cheap Zosi 4 camera CCTV+ DVR system (without hard drive) for $105 with good reviews on Amazon (I know many reviews on Amazon are fake).
But it's not wireless and I need wireless access to my phone to view a live video stream and recorded video. So the question is, can I connect the NVR to a wireless router and get the live local feed and access to recorded video on my phone over LAN wifi?.
Also, it's not PoE, so I'm assuming I need an outlet to plug them into, or is there a PoE adapter I can use for it?
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