Prosper comes from the prefix pro- and the latin root sper, which means "hope." another word from the same root is despair, which means "deep sadness." explain how despair might come from the sper root.

The Latin root word -sper is found in many synonymous words; desperado, desperate, prosper, prosperity, prosperous i.e.

Desperate means; the feeling, showing, or involving a hopeless sense that a situation is so bad as to be impossible to deal with.
Desperate has the root word -sper.

Now despair means; the complete loss or absence of hope. Desperate, and despair both have the de- prefix, used to indicate privation, removal, and separation. Both words meaning lack of hope or absence of it.

The word prosper means to succeed in material terms. Financially stable i.e. For someone to have despair, means they have no hope of being prosperous.

Connecting it all together you can see that they all go hand in hand one way or another de- meaning to remove, sper meaning hope. Desperate meaning you have no hope. Despair means you you also have lack of hope. etc.

Hope this helped, if you have any questions please ask!