a) by constantly defeating French armies in Spain, draining the resources of the Empire which would have been useful elsewhere.
b) by winning at Waterloo and finally seeing Napoleon off to St. Helena.
There is a mistake in your question:
Both Duke of Wellington and Admiral Nelson were English, and neither one defeated the other!
But both of them are famous for defeating Napoleon at the Battle of Trafalgar in 1805 and Battle of Waterloo in 1815.
Admiral Nelson was killed in Trafalgar, but he still managed to defeat French Admiral Villeneuve who was in charge of the Franco-Spanish fleet that was based in Cadiz.
Nelson won thanks to his naval skills and naval supremacy. Although the French fleet had more vessels, quite a number of them didn't participate in the battle.
Napoleon was defeated by Wellington because he failed to receive reinforcements from his officer Crouchy, who could have cleared the French army away by attacking the flanks of the Allies.
Because he was British and a better strategist and soldier than Napoleon. The duke which then was know as Wesseley knew that napoleons cavalry played a major role in battles, so he Waited for Napoleons cavalry to come to flank his regiments, then flanked his Calvary thus surrounding the Napoleonic army.
Or:
Napoleon's Army was essentially untrained. The troops were armed with muskets while Wellington had many units armed with the longer ranged and more accurate rifle. The UK soldier was highly trained, and stood his ground in every engagement. He was resolute and determined. He would not yield an inch to the French Army.
Arthur Wellesley was the Duke of Wellington around Napoleon's time.
He defeated Napoleon at Waterloo.
Arthur Wellesley, 1st Duke of Wellington"Duke of Wellington" and "The Iron Duke". .
Napoleon Bonaparte lived from the 15th August 1769 to the 5th May 1821. Field Marshal Arthur Wellesley, 1st Duke of Wellington lived from the 1st May 1769 to the 14th September 1852.
The French Revolution ended with the creation of the French Consulate by coup in 1799 in which Napoleon became the First Consul of France. The legal government of the Directory ceased to exist.
Napoleon
The Duke of Wellington.
The Duke of Wellington with plenty of Prussian help.
The French Emperor, Napoleon Bonaparte I.
Napoleon Bonaparte suffered defeat at the hands of the Duke of Wellington and the British army
Arthur Wellesley was the Duke of Wellington around Napoleon's time.
They fought the Battle of Waterloo om 18 June 1815.
Napoleon was defeat by Arthur Wellesley, the 1st Duke of Wellington.
The English general was the Duke of Wellington. The battle of Waterloo took place during the reign of George IV.
In a nutshell, Waterloo, in Belgium, was the location of Napoleon's defeat (French), by the Duke of Wellington (British). It meant the end of Napoleon's rule and domination over Europe.
Waterloo in Belgium
Waterloo in Belgium