I'm guessing that you don't because of the question but if you have a buddy who does, give him a visit. If your rifle is a bolt action, you can do a quick bore sight without tools, by popping the bolt out, placing it in a gun vise or somewhat steady cradle of some sort, and look down the bore then adjust the scope to match where the bore is looking. Also before you get too carried away make sure the mount and ring screws are snug, overtightening them is a bad thing but hold the screwdriver with just your thumb and 2 fingers and that's about the right pressure to turn the screw with for sufficiently tight.
Having it too tight can torque your scope leading to problems. If everything's tight and bore sighting's not an option, then bring the target in way close like you thought of. A lot of rings are set on the mounts by a large screw on the side, which will provide some windage adjustment.
I'm willing to bet that one of these is a little out of adjustment, setting your scope at a slight horizontal angle.
It can be the scope for sure; my Bosch and Lomb costs over $200 and it was pretty accurate at the first 100 yards when testing. It is sometimes normal for this to happen depending on what it is you're shooting. If you do not have integrated sights on your gun, this will be a little more difficult to figure out, but you are doing the right thing and shooting at 10 yards.
I have a friend that has placed a scope on a Bushmaster AR-15 and he had to do the same thing that you're about to do and sight the gun in from increments of 10- it is time consuming, but it is well worth it when you're accurate at 250 yrds. Check your scope mounts, you may have an issue with them. Lastly, a 2' x 1' piece of paper for the first shot(s) is not giving a lot of space to figure it out, I use a 4' x 8' piece of plywood when I am sighting in.
You may also want to invest in a bi-pod or some sand bags to assist in sighting in, it helps for some folks. If all else fails, take the scope back and invest in a premium brand, well worth it in the long run. When my refund check comes in, I am going to get a scope with the built in Trijicon illuminated cross hairs, they are freaking sweet.
Good Luck!
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