## Definition 3.5.7: Exponential Sequence

**Exponential Sequence**: Converges to the exponential function

*e*for any real number

^{x}= exp(x)*x*.

### Proof:

We will use a simple substitution to prove this. LetThen we havex/n = 1/u, or equivalently,n = u x

But the term inside the square brackets is Euler's sequence, which converges to Euler's number

*e*. Hence, the whole expression converges to

*e*, as required.

^{x}In fact, we have used a property relating to functions to make this proof work correctly. What is that property ?

If we did not want to use functions, we could first prove the statement for
*x* being an integer. Then we could expand it to rational
numbers, and then, approximating *x* by rational number, we
could prove the final result.