We need to find ways of enhancing the potency of existing antibiotics, and, with this in mind, we begin with an unusual question: how low can antibiotic dosages be and yet bacterial clearance still be observed? Seeking to optimise the simultaneous use of two antibiotics, we use the minimal dose at which clearance is observed in an in vitro experimental model of antibiotic treatment as a criterion to distinguish the best and worst treatments of a bacterium, Escherichia coli. Our aim is to compare a combination treatment consisting of two synergistic antibiotics to so-called sequential treatments in which the choice of antibiotic to administer can change with each round of treatment. Using mathematical predictions validated by the E. coli treatment model, we show that clearance of the bacterium can be achieved using sequential treatments at antibiotic dosages so low that the equivalent two-drug combination treatments are ineffective. Seeking to treat the bacterium in testing circumstances, we purposefully study an E. coli strain that has a multidrug pump encoded in its chromosome that effluxes both antibiotics. Genomic amplifications that increase the number of pumps expressed per cell can cause the failure of high-dose combination treatments, yet, as we show, sequentially treated populations can still collapse. However, dual resistance due to the pump means that the antibiotics must be carefully deployed and not all sublethal sequential treatments succeed. A screen of 136 96-h-long sequential treatments determined five of these that could clear the bacterium at sublethal dosages in all replicate populations, even though none had done so by 24 h. These successes can be attributed to a collateral sensitivity whereby cross-resistance due to the duplicated pump proves insufficient to stop a reduction in E. coli growth rate following drug exchanges, a reduction that proves large enough for appropriately chosen drug switches to clear the bacterium.