What finally led to the surrender of Pemberton and his troops at Vicksburg?

The siege of vicksberg

Vicksburg was the last major Confederate stronghold on the Mississippi River; therefore, capturing it completed the second part of the Northern strategy, the Anaconda Plan. When two major assaults (May 19 and 22, 1863) against the Confederate fortifications were repulsed with heavy casualties, Grant decided to besiege the city beginning on May 25. With no reinforcement, supplies nearly gone, and after holding out for more than forty days, the garrison finally surrendered on July 4.