To prove the convergence of an inner product space

Question

I am trying to prove that $\mathbb R^2$ is a Hilbert space (complete inner product space). We know that $\mathbb R^2$ is an inner product space (its inner product on $\mathbb R^2$ is simply a dot product). I am having trouble with proving that $\mathbb R^2$ is complete, i.e. every Cauchy sequence in $\mathbb R^2$ converges to an element in $\mathbb R^2$. Any help? [or] Please answer if $\mathbb R^2$ is a finite dimensional space. I reckon yes, but I am not sure how.

Answer 1

First of all $\mathbb{R}^2$ is an inner product space by the inner product $\langle a,b\rangle=a \cdot b$, for all $a, b\in\mathbb{R}^2$.

Let $\{(x_n,y_n):n\in\mathbb{N}\}$ be a Cauchy sequence in $\mathbb{R}^2$ then $\|(x_n,y_n)-(x_m,y_m)\|\rightarrow 0$ as $m,n\rightarrow\infty$

$\iff\|(x_n-x_m,y_n-y_m)\|\rightarrow 0$ as $m,n\rightarrow\infty$

$\iff x_n-x_m\rightarrow 0,y_n-y_m\rightarrow 0$ as $m,n\rightarrow\infty$

$\{x_n:n\in\mathbb{N}\}$ and $\{y_n:n\in\mathbb{N}\}$ are Cauchy sequence in $\mathbb{R}$, $\mathbb{R}$ is complete so $\exists x,y\in\mathbb{R}$ s.t. $x_n\rightarrow x$ and $y_n\rightarrow y$ as $n\rightarrow\infty$

Now, $\|(x_n,y_n)-(x,y)\|=\|(x_n-x,y_n-y)\|\rightarrow\ 0$ as $n\rightarrow\infty$. Hence, $\mathbb{R}^2$ is complete.