Cardinality of Linearly Independent Set is No Greater than Dimension

Theorem
Let $G$ be a vector space of $n$ dimensions.

Let $H$ be a linearly independent subset of $G$.

$H$ has at most $n$ elements.

Proof
Let $H$ be a linearly independent subset of $G$.

By definition of dimension of vector space, $G$ has a basis with exactly $n$ elements.

By Sufficient Conditions for Basis of Finite Dimensional Vector Space, $B$ is a generator for $G$.

Then by Size of Linearly Independent Subset is at Most Size of Finite Generator, $H$ has at most $n$ elements.