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When I was travelling in India in 1984 for 6 weeks, I didn't want to get completely out of shape so I improvised to the max. For some reason there was a bucket in the shower, so I filled the bucket with water and did curls with a bucket full of water.
First of all pushups do not work biceps...you could probably not have biceps and still manage to do pushups.
Elastic bands are great and you can take them anywhere...tons you can do for biceps. Any motion you do where the resistance is pulling your arm upward and flexing your bicep is going to be working that muscle. Get bands that you can lock inbetween a door and slide it to the bottom so you can pull upwards like you are curling. Also a lot of motions that use your back also use your biceps...so if you wanted to have the elastic bands in the middle of the door and pull the band towards you it would work biceps and upper back.
Pull ups are great and you want to aim for being able to do them. Once you get to the point where you can do sets of even 6 pull ups it means you are strong enough to use your own body weight...thats huge for most people. It will tone the muscle as well as build strength but it wont add any mass unless you put a chain around your waist and start adding weight to the pullups (I'm just now starting to work on this myself).
Why cant you use dumbells? If you arent trying to add mass you and you're a female (judging by your screen name) you really only need a couple of sets of light weights like 5 pounds and maybe 8 or 10 pounders. This isnt a lot of money...heck you can just buy one dumbell if you're that concerned and rotate arms when you exercise. High repetitions and low weight will tone muscle and burn fat as well as add a minimal amount of strength. I bet you can pickup a 5,8,10,12 pound set of dumbells for under 100 bucks...and you dont need anything fancy but the dumbells to do about a dozen different exercises for every muscle in your upper body. You will want a bit more weight for back and chest though in most cases.
So if you're looking for real cheap and ease of use (as in anywhere) buy a set of elastic bands on something like Amazon for about 40 bucks. You can use them anywhere there is a door. Then find ways to pull on them that work your biceps. If you have more money like 100 dollars pickup some sets of low weight dumbells...even if you have to go with one for each weight it will allow more free range of motion and creativity with exercise than bands.
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