# Apostol proof of linear independence of n exponentials

This following proof is from Apostol's Calculus Volume 2.

Example 7
If $$a_1,...,a_n$$ are distinct real numbers, the $$n$$ exponential functions
$$u_1(x)=e^{a_1x},...,u_n(x)=e^{a_nx}$$
are independent. We can prove this by induction on $$n$$. The result holds trivially when $$n=1$$. Therefore, assume it is true for $$n-1$$ exponential functions and consider scalars $$c_1,...,c_n$$ such that
$$\sum_{k=1}^{n}c_ke^{a_kx}=0$$.     (1.2)
Let $$a_M$$ be the largest of the $$n$$ numbers $$a_1,...,a_n$$. Multiplying both members of (1.2) by $$e^{-a_Mx}$$, we obtain
$$\sum_{k=1}^{n}c_ke^{(a_k-a_M)x}=0$$.     (1.3)
If $$k \not= M$$, the number $$a_k-a_M$$ is negative. Therefore, when $$x \rightarrow +\infty$$ in Equation (1.3), each term with $$k \not= M$$ tends to zero and we find that $$c_M=0$$. Deleting the $$M$$th term from (1.2) and applying the induction hypothesis, we find that each of the remaining $$n-1$$ coefficients $$c_k$$ is zero.

I understand what is happening up to and including when he states that "If $$k \not= M$$, the number $$a_k-a_M$$ is negative." However, then I am confused as to why he is considering the limit as $$x$$ tends to infinity. it seems arbitrary. And also, how does applying the induction hypothesis make sense? From what I have learned, the inductive step is proving that if $$k$$ is true $$k+1$$ is true, but I don't see how he is doing that.

Our assumption is that $$\sum_{k=1}^{n}c_ke^{a_kx}=0$$ for all $$x\in \Bbb R$$. So in particular, it should hold as $$x$$ tends to $$\infty$$. When $$x\to\infty$$, $$e^{(a_k-a_M)x}\to 0$$ for all $$k\neq M$$. Then the only term left on the LHS is $$c_M$$, which is equal to zero on the RHS. So we have shown that $$c_M =0$$. Now we have only $$n-1$$ terms on the LHS. By induction assumption, we already know that $$n-1$$ terms of $$e^{a_kx}$$ are linearly independent. Hence $$c_k=0$$ for all $$1\leq k\leq n$$.
• But then is he not proving it only for $x \rightarrow \infty$ instead of for all $x \in R$? Jun 23, 2019 at 2:26
• @JohnArg Our assumption is that $\sum_{k=1}^{n}c_ke^{a_kx}=0$ for all $x\in \Bbb R$. Note that the coefficients are independent of $x$. So the same coefficients should be there for $x$ tending to infinity also.
Note that $$u_1(x),u_2(x),...,u_n(x)$$ are functions with domains $$(-\infty,\infty)$$ so it makes perfect sense to fond the limit as $$x\to \infty$$
Regarding your induction question note that once one of the coefficients becomes zero the case on $$n$$ coefficients goes back to the case of $$n-1$$ coefficients hence the induction hypothesis kicks in.