"Promises made easily are bound to be easily broken" corresponds to three famous sayings: "A light promise will lead to a reticence", "A broken promise will not be established", and "A small faith will lead to a big faith."
"A light promise will lead to a lack of words" comes from Chapter 63 of "Laozi", which means that a promise made easily must lack credibility.
"Breaking trust and not standing" comes from "Zuo Zhuan·Xiangong Twenty-Two Years", which means that people who do not keep their trust cannot gain a foothold in society. It is extended to the inability to gain a foothold in society and the inability to gain recognition from others.
"When small faith is established, great faith is established" comes from "Han Feizi·Wai Chu Shuo, Part 1", which means that only when small integrity is established, great integrity can be established.