Originally Posted by
coquitao Let its limit be L. The very definition of convergence implies the existence of a natural number N such that $\displaystyle |a_{n}-L|<1$ for every $\displaystyle n \geq N$. The former inequality implies in turn that $\displaystyle |a_{n}| < 1+|L|$ for every $\displaystyle n \geq N.$
Hence, starting from the N-th element of the sequence you can place them all inside of a disk of radius 1+|L| (and centered at the origin). it's clear that up to this moment some elements of the sequence might remain outside of this disk, but since it's only a finite number of them, we are done, right?