Every convergent sequence is bounded, proof

Suppose we have a sequence a_n which converges to L.

Let e > 0, then there is an integer N > 0 such that for all n > N we have,

|a_n - L| < 1. Why 1? Well 1 is a number, we can use any positive number here.

Then adding and subtracting L and using the triangle inequality we have,

|a_n| = |a_n - L + L| <= |a_n - L| + L  < 1 + L for all n > N.


Set M = max{a_1, a_2, ..., a_(n-1), 1 + L}

Then, for ALL positive integers n we have,

|a_n| <= M.

This shows the sequence is bounded.