Proving size of two finite sets is equal to the size of their union I am struggling to understand how to prove the following for ALL numbers. 
Show that if S₁ and S₂ are finite sets with |S₁| = n and |S₂| = m, then |S₁ U S₂| <= n + m.

It is easy for me to show this by creating two finite sets and going through the computations, but I am struggling to understand how to show this in general terms, or for ALL numbers. 
Here is how I show it with sets that I created. 
S₁= {1, 4, 5, 7}
|S₁| = 4 
S₂ = {4, 6, 8}
|S₂| = 3
S₁ ∪ S₂  = {1, 4, 5, 6, 7, 8}
|S₁ ∪ S₂| = 6
6 <= (4 + 3)
6 <= 7

I am not seeing how to translate this to general terms. 

It is a standard theorem (see, e.g. wiki), that $$|S_1|+|S_2|=|S_1\cup S_2|+|S_1\cap S_2|$$
We rearrange as $$|S_1\cup S_2|=|S_1|+|S_2|-|S_1\cap S_2|\le |S_1|+|S_2|$$
