My bike shifting won’t surpass 6

It sounds like your derailleur cable has a bit of slack in it. You will find this video useful, its easier to give you this link with a lot of good footage than try to explain it myself.

Lets assume that you can change into the lowest gear (largest sprocket, gear 1 indicated on the shifter), but as you change to higher gears the derailleur stops on the sixth sprocket, when the shifter is moved to 7, 8 or 9.

The shifter lets cable out as you change to higher gears and the derailleur spring pulls the cage outwards. Either there is something jamming the derailleur directly, or the cable is hanging up and preventing the derailleur from moving beyond the sixth sprocket.

Shift to 9 on the shifter, detach the cable from the derailleur by undoing the pinch bolt. If you can make the derailleur shift to all the sprockets by pushing on it while pedally manually then it's not jammed. (Be careful not to catch your fingers in the chain).

Try pulling the cable while changing up and down with the shifter, if you feel resistance in the cable while shifting up to 9 then possible the cable is jamming. You can try pulling the cable from the housing, cleaning it and re-threading it, or replacing it. You may also find the housing needs replacing.

When you put at all back together, follow the proper procedure for setting limits and indexing.

This answer assumes that you're dealing with a 2x9drivetrain and that your gear shifter reads 1 when the chain is on the large rear sprocket and when shifter reads 9, the chain SHOULD be on the smallest rear sprocket. In addition, assumes mechanical system that relies on inner cable tension and movement to move (shift) the derailleur.

Based on your description, I read that your chain will not move onto the smaller sprockets--or high gear side--of the cassette even though the shift indicator reports it should be. The heart of the problem then, is that the rear derailleur is unable to move the chain outbound past the 4th smallest sprocket. A couple reasons for this would be incorrectly set limit screw (specifically the "H" limit screw) or there is excess friction in the cable system such that the derailleur's spring tension is unable to overcome it and move derailleur further outside onto the smaller, high gear sprockets. A third, more problematic and expensive, possibility is that due to excessive wear or breakage, the spring tension is too weak for the derailleur to function properly. The most common cause of shifting problems is related to cable tension...too much or too little.

To start your diagnosis, completely unhook the inner cable from the derailleur's pinch bolt. Does anything happen? In this state, the derailleur's springs are totally relaxed and the derailleur should be as far outside as the high limit screw will let it. The proper setting of which is one that brings the derailleur's upper jockey wheel right under the outside edge of the small cog. So from behind the bike, check where the jockey wheel sits related to the cassette cogs. Adjust the "H" screw to get the jockey wheel to line up with the small sprocket. Clockwise turns of the screw moves the jockey wheel further inboard, while loosening (counterCW) the screw causes the jockey wheel to move outward. The degree of movement is small--a few millimeters, just to get good alignment. You've now set the High limit of the rear derailleur. Try moving the derailleur by hand to the inside. Then allow the spring tension to return it back to it's outside position. With the chain still wrapped, you must turn the pedals when you move the derailleur. Does the derailleur move the chain across the entire cassette? And then does it's extended spring bring it back down to inline with the small sprocket when you release the deraileur (keep rotating the crankarms)? If this seems to be the case that the rear derailleur can be manipulated by hand across the full range of the cassette, and it can (without assistance) bring the chain back down to the small cog, your rear derailleur is probably working correctly.

You should check the low limit screw at this time: while turning the crank, push the rear derailleur as far to the inside as you can. The chain will climb the cassette. Be careful when getting to the large sprocket. If the low limit screw is too loose, the derailleur will overshoot the line of the large sprocket and send the chain (& possibly itself) into the spokes: not good. If the chain seems to want to derail past the large sprocket as you manually push the derailleur inside (while slowly rotating the crank), turn the low limit screw in (clockwise). If you've reached the limit of the derailleur's inside movement and the chain hasn't climbed onto the larger sprocket, loosen the low limit screw about 1/8 of a turn until the chain reaches the large sprocket. The best low limit setting is when the upper jockey wheel of the deraileur is right under the large sprocket (inline) AND the derailleur cannot be pushed further inside. When we've hooked the cable up and are working on the indexing of the gears, it may be necessary to adjust the limit screws A VERY LITTLE BIT. Care must be taken with the low limit screw to prevent derailment into the rear wheel.

If the derailleur assessment checks out, move on to checking the cable and the action of the shifter upon it. The most common cause of shifting issues is excess friction in this system. Rust, frayed or kinked cable, worn ferrules, too tight of a turn of the cable, contamination of the inside of the outer cable with dirt: all are common causes of the cable system dysfunction. Inspect the cable along it's length. The ends of each section of outer cable should be fully seated into the frame's cable stops. The sections should have gentle, sweeping flows between stops without sharp bends or kinks, the exposed inner cable should be free of corrosion, no breaks or kinks in the strands. Pay particular attention to the several millimeters around each cable stop where the inner cable moves in and out of the outer housing. ANY disfigurement of the inner cable here can cause problems. While the cableing is still in place in their stops, take hold of the inner cable near where the pinch bolt held it. Putting some gentle tension on the cable, click the shifter through the range of gears. You should just feel the smooth movement of the inner cable--inward as you shift down (the numbers decreasing on your shift indicator). When connected this action moves the derailleur inbound, causing the chain to climb the cassette sprockets, small to large. Next, shift the other way--indicator numbers are increasing. (Pay close attention at this step since it has the highest probability for a problem based on your symptoms. Especially if you've found the derailleur's high limit setting ok as well as it's spring action). When you shift while putting gentle tension by hand on the inner cable, the cable should slide further out the end a few millimeters each shift. You shouldn't be able to feel anything but the swift, smooth movement of those few mm's. Nor should you have to feel like you had to "pull" or increase your finger tension to get the cable to move. Starting with the slightest tension on the cable, a shift should cause your hand to slip back a few mm toward the direction of tension. Any little hiccup or hesitation of the cable movement is indicative of a problem. You can further isolate where the problem lies by moving up past a section of housing and again shifting while holding on to the inner cable.

Here's a few instructional videos relevant to your problem. It's always a good idea to change out your shift cables (inner AND outer) every so often. For around $20 USD, you get everything you need to do both shifters on a bike. Rear Derailleur Adjustment; Troubleshooting the Rear Derailleur; Cable Cutting and Sizing