Show that the space of infinite sequence of real numbers is infinite dimensional.
The solution is the space V of infinite sequence is infinite dimensional.
Assume that the opposite of that the space V of infinite sequences is finite dimensional with as follows.
Consider an example of n+1 linearly independent infinite sequence as follows.
This contradicts the fact that n is the largest possible dimension for an n-dimensional space that m=n.
Therefore, the assumption is wrong.
Thus, the space V of infinite sequence is infinite dimensional.
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