If you want a mathematical proof, we can use the precise definition of the limit (epsilon-delta).
For a really simple explanation, say that you have a near infinitely long paper, and you plot (n, a_n) on it.
a_n is said to converge to L, when, if you draw two lines close to y=L, say, y=L - 1 and L + 1, and you look to your right, at some point all values of a_n bigger than a certain n are plotted within those lines. And even if you close the gap, like L - 0.1 and L + 0.1, if you look further right there is a point where a_n all fits within those lines.
From here, say that you picked any number of values and changed them to 500. Let's say the biggest one is n = m, hoping that you will not be able to repeat the process I mentioned and the limit will change. Unfortunately, however, you can always look further right than n = m+1, and find an n where it still fits within those lines. So the limit won't change.