The integral statement is most generally this:

For a based CW space $X$ one has a suspension map $E: X\to \Omega \Sigma X$ and
a Hopf invariant $H: \Omega \Sigma X \to \Omega \Sigma (X\wedge X)$ (construction outlined below).

The composite
$H\circ E$ is canonically null and the sequence
$$
X \overset E \to \Omega \Sigma X \overset H \to  \Omega \Sigma (X \wedge X)
$$
is a homotopy fiber sequence in a range, roughly $3r$, where $r$ is the  connectivity of $X$.
Your result will follow from this easily (take $X = S^n$).

The map $H$ is often constructed as follows: let $JX$ be the reduced free monoid on the points of $X$. Using say the Moore loops model for $\Omega \Sigma X$, one can construct a monoid homomorphism $JX \to \Omega \Sigma X$ extends the map $X$ using the universal property of the free monoid. A homology calculation shows that this map his a weak equivalence (James did this calculation when $X$ is a sphere).

Finally, the proof the above sequence is a fibration in the range roughly thrice the connectivity of $X$ can be deduced from the following four facts:

1) $J_2 X \to JX$ is $3r$-connected, where $J_2X = X \cup (X \times X)$ is filtration two.
 
2) The quotient $J_2X/X$ is $X\wedge X$, so we have a cofiber sequence
$$
X \to J_2 X \overset q\to X \wedge X
$$

3) By the Blakers-Massey Theorem, the above sequence is a homotopy fibration in
the range roughly $3r$.

4) The diagram
$$
J_2X \,\, \overset q\to X\wedge X
$$
$$
\downarrow \qquad \quad\,\,\downarrow 
$$
$$
JX \underset H\to \Omega \Sigma (X\wedge X)
$$
is commutative, where the left vertical maps is the inclusion and the right one is the suspension map for $X\wedge X$.