Prim's Algorithm produces Minimum Spanning Tree

Theorem
Prim's Algorithm always produces a minimum spanning tree.

Proof
Suppose that Prim's Algorithm produces a tree $T$.

Let there exist another spanning tree $S$ with a smaller total weight.

Let $e$ be an edge of smallest weight which lies in $T$ but not $S$.

If we add $e$ to $S$, we obtain a cycle, from Equivalent Definitions for Finite Tree.

This cycle contains an edge $e'$ which is in $S$ but not $T$, otherwise $T$ would not be a tree.

So, we replace $e'$ in $S$ with $e$ from $T$, and obtain a new spanning tree $S'$.

From the method of construction of $T$, it follows that the weight of $e$ can not exceed that of $e'$.

So the total weight of $S'$ does not exceed the total weight of $T$.

Also, $S'$ has one more edge in common with $T$ than $S$ has.

We repeat the above procedure, and repeatedly change edges of $S$ for edges of $T$, and each time the weight of the intermediate graph does not exceed that of $T$.

Thus the weight of $T$ does not exceed that of $S$, contradicting the definition of $S$.

Hence $T$ must be a minimum spanning tree.