What event prevented an outbreak of war between the united states and mexico in 1917 what did the united states prove by its actions in mexico.

War never broke out between the US and Mexico in 1917. British intelligence intercepted a telegram from Arthur Zimmerman of Germany destined for Mexico City officials in Mexico. This telegram (known as the Zimmerman telegram) was dialogue encouraging Mexico to attack the United States if the US entered the war and ended up declaring war on Germany due to its resumption of unrestricted submarine warfare in the Atlantic Ocean. The telegram never got to Mexico, but it did reach the office of US President, Woodrow Wilson. Wilson declared war against the Central Powers shortly after the telegram was leaked to him. No war broke out between the US & Mexico, but there certainly was tension. 
For your second question, I'm assuming your talking about the assassination of Francis Ferdinand leading to WWI? The heir to the Austro-Hungarian throne, Francis Ferdinand, was visiting Sarajevo, Bosnia in June of 1914, when he was assassinated by a Serbian terrorist group called The Black Hand. Outraged, Austria-Hungary sent an ultimatum to Serbia, which Serbia did not respond to. With the full backing of its ally, Germany, Austria-Hungary then declared war on Serbia. This prompted Russia to mobilize its army in support of Serbia, which prompted Germany to declare war on Russia and then France. Germany had been chomping at the bit to flex its military might and expand its empire and saw this as the perfect opportunity. As German forces went through Belgium (to get to France), Great Britain was forced to declare war on Germany in support of Belgium's neutrality agreement that had been violated by the Germans. 

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