Last checked at 
* FTSE 100 up 1.7 pct
* Sterling sinks further after May lays down gauntlet to EU
* Banks and miners drive gains
* Just Eat tumbles on report Uber in talks to buy Deliveroo(Updates prices, adds details, quotes)
By Helen Reid
LONDON, Sept 21 (Reuters) - A slide in sterling after aspeech by Prime Minister Theresa May helped boost Britain'sexporter-heavy FTSE 100 to a 2 1/2 week high, cementing athree-day rally driven by banks, miners and oil stocks asinvestors' fears of a trade war faded.
On the corporate front, Just Eat shares tumbled as Uber'sreported advances to Deliveroo triggered fears of risingcompetition in the food delivery industry.
The FTSE 100 ended the day up 1.7 percent, extendingearly gains as developments on the Brexit front sank sterling,supporting the index whose constituents mainly earn outside theUK.
The pound tumbled two cents against the dollar, set for itsbiggest daily drop this year, after Prime Minister Theresa Maysaid Brexit talks with the European Union had reached an impasseand called for new proposals.
"The near-term path of UK stocks and sterling is highlydependent on the Brexit newsflow and as a result we maintain anunderweight position to both, instead favouring themes andgeographies with more economic certainty," said Edward Park,investment director at Brooks Macdonald.
The FTSE 100 had its third day of gains in a row - itslongest winning streak since mid-August. The index has beenstruggling for momentum as Brexit negotiations wear on.
A rare faller in a strong market, food delivery company JustEat tumbled as much as 7.5 percent after a report thatUber was in early talks to acquire its rival Deliveroo.
"Uber’s deep pockets combined with an acquisition puts a lotof competitive pressure on the other food delivery companiesover the medium term," said Brooks Macdonald's Park.
European benchmarks all climbed as investors piled intostocks in relief that U.S.-China tariffs announced this weekwere lower than feared.
Investors also believed their impact on global growth wouldbe muted, save for a slight increase in inflation - which is anegative for bond markets but a positive for equity markets.
Overall, banks were a top driver of FTSE 100 gains.
Heavyweight banking stock HSBC was the biggestboost to the index, up 1.3 percent, driven higher by a spike inthe Hong Kong dollar overnight caused by expectations of a risein bank lending rates and tightness in cash supplies.
Standard Chartered, which also has significantexposure to Asia, also gained 1.8 percent, among top FTSErisers.
Mining stocks Glencore, Anglo American, RioTinto and BHP Billiton rose 2 to 3 percent asmetals prices added to this week's gains.
Copper hit its highest in six weeks and was set for itsbiggest weekly climb in 15 weeks as investors expected tradetariffs to have less impact on global growth than some hadfeared.
"Our economists expect trade disputes to linger with noresolution in sight, but China plans to offset the headwindswith infrastructure spending and liquidity support," GoldmanSachs analysts wrote, adding that they see room for copper torise further.
Oil majors BP and Royal Dutch Shell alsospurred the FTSE 100 higher as crude prices rose ahead of anOPEC meeting on Sunday.
Shares in industrial group Smiths fell 4.9 percent,the top FTSE 100 fallers.
Smiths reported full-year profit which missed analysts'estimates due to short-term problems at its medical unit due tothe loss of some product certifications under new Europeanregulations.
(Reporting by Helen Reid; Editing by Andrew Roche)
