LONDON, Sept 22 (Reuters) - The pound fell for a third day on Wednesday, driven lower by the strength of the dollar, which got a lift from investors pricing in a higher chance of a string of US rate rises over the coming months, even as the oil price trades below $100 a barrel.

Sterling was around $1.328, down 0.45% on the day, even with oil futures dropping below $99 again, as investors clung on to evidence that supply from the Middle East was slowly improving, while optimism grew over a possible diplomatic breakthrough between the US and Iran. 

Growth in British business activity cooled this month and inflation pressure built, a survey showed on Wednesday, an awkward backdrop for finance minister John Healey ahead of his first budget in October.

The S&P Global UK Services Purchasing Managers' Index (PMI) fell in September to 51.7 from 52.5 in August, a three-month low, according to "flash" or preliminary data. A Reuters poll of economists had pointed to a reading of 52.0.

"Growth in Britain’s economy has been remarkably resilient so far this year, though we think that a slowdown is almost inevitable during the remainder of the year - energy costs have risen, borrowing costs are up, the jobs market continues to weaken and political uncertainty looks set to rear its ugly head again as we approach budget day next month," Matthew Ryan, head of market strategy at Ebury, said.

"We expect this to keep sterling under pressure in the near-term, though we do contend that sterling appears a bit oversold at current levels."

Money markets show traders expect UK rates to be around a full percentage point above the current rate of 3.75% this time next year, which would imply four quarter-point hikes between now and then.

They assign roughly a 65% chance of a rate rise at the BoE's November meeting, which falls right after the Autumn Budget, with a December hike seen as a done deal.     

(Reporting by Amanda Cooper; Editing by Andrew Cawthorne)

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