## 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
$$C=sup_nintvarphi(x)dF_n(x)
then $$F_n$$ is tight.

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