The statement for which I am trying to prove or provide a counter example is the following:
Let $V$ be a vector space with dim$V=n$, and let $B$ be a basis for $V$. If $U$ is any subpsace of $V$ with $1\le dim U\le n-1 $, then $B$ contains a basis for $U$
From what I got out of it it's saying that, given a vector space $V$ with dimension $n$, there exists a subspace $U$ in $V$ such that the basis of $U$ is contained in the basis of $V$, as long as the subspace of $V$ is between the dimensions of 1 and $n-1$.
My problem with the question isn't understanding it but rather finding where to start to either prove or disprove it. On my initial reading of the statement, this seems like it would be true since $U$ is a part of $V$ and therefore the basis that makes $V$ should also be able to create $U$.
I am concerned about the final part of the statement saying that $1\le dimU\le n-1$. I understand that if $U$ has a dimension of 0 then it has no vectors in it's basis and if it has $n$ vectors then the basis of $U$ is just equal to the basis of $V$ meaning $U=V$.
My way of proving the validity of this statement would be by saying the following:
If $U$ is a subspace of $V$ then contains vectors found in $V$. The basis of a vector space is a set of vectors which can be manipulated with addition and scalar multiplication to form any vector found within the vector space. Given that the $U$ consists of vectors found in $V$ and the basis of $V$ can create any vector found in $V$, it is possible to use a set of vectors from the basis of $V$ to construct a basis for $U$ since $U$ is just a subset of vectors found in $V$.
I'm not sure if this counts as proving the following statement and also I was not able to think of any counter examples but the statement could be false and I might have missed it. If anyone would like to point me in the right direction here i would greatly appreciate it.
While @dxiv's comment answers the question, I want to provide pretty same reasoning.
So, you know $|B| = n$, then any subset of $B$ is a basis to some subspace, so you have at most $2^n$ possible subspaces, that could be formed by subsets of $B$.
This leads you to immediate conclusion, that the premise of the question is wrong (at least above infinite field) for $n \geq 2$, just by observing there are infinitely many $span\{b_1 + a \cdot b_2\}$ for the first two vectors in basis.