Theorem 3.2.14 (Probability: Theory and Examples, Durrett, 5th ed.)

The proof of the theorem is a single line and thus seemingly straightforward, yet I cannot see how to justify it. I tried to rearrange some of the pieces, but that approach failed to improve my understanding. Any help in justifying the statement would be greatly appreciated.

$textbf{Theorem 3.2.14.}$
If there is a $varphige0$ so that $varphi(x)tointfy$ as $|x|toinfty$ and
$$
C=sup_nintvarphi(x)dF_n(x)<infty,
$$

then $F_n$ is tight.

$textbf{Proof.}$
$$
1-F_n(M)+F_n(-M)lefrac{C}{inf_{|x|ge M}varphi(x)}.
$$