Is finding a $800-$950 single or 1 bedroom possible in Los Angeles?

My limit is $1000 but I prefer not to go there. I don't care if it's a single unit, even though I would prefer a 1 bedroom. It's me, my boyfriend and 1 year old son.

We live in the valley and have to move and we're hoping to find something in LA edit(between downtown and the valley!) What's your experience?

Also, do you recommend searching on foot rather than online ads? Thanks a bunch!

Edit: thanks everyone! I know it's a crazy request but everyone irl tells me it's doable so I thought I would reach out online for any extra information I might be missing. I got a bunch of areas to check out, thanks to you folks.