Question about Lemma 1 in "A.2. Proof of Theorem 1"

[phucdoitoan]

The paper states that Lemma 1 is taken from (Chung, 2007).
However, the closest thing I found in (Chung, 2007) is Lemma 3 ( inequalities (51), (52) ), in which the upper bound contains (beta_k)^2/8 not (lambda_G)^2/2.
I tried to derives Lemma 1 in the paper from Lemma 3 in (Chung, 2007) but have not succeeded.

I would really appreciate if you can provide a proof or point out any sources that lead to the proof of Lemma 1 in the paper.
Thank you!.

(Chung, 2007): Four proofs for the cheeger inequality and graph partition algorithms, ICCM 2007