# Assorted Analysis Proofs

This post lists assorted proofs from **Analysis**, without any particular theme.

#### Prove that if \(S\) is open, \(S'\) is closed.

**Proof:**

We claim that if \(S\) is open, \(S'\) is closed.
Thus, we’d like to prove that for a sequence \((x_k) \in S'\):

\[\text{lim}_{k \rightarrow \infty} (x_k)= x_0 \in S'\]
We will prove this by contradiction.

Assume that \(\require{cancel} x_0 \cancel{\in} S'\). Then, \(x_0 \in S\).

Since \(S\) is open, there exists an \(r>0\), such that \(d(x_0,p)<r\); that is, there exists an \(r\)-neighbourhood around \(x_0\) in \(S\).

Choose \(\epsilon<r\), then there exists \(N \in \mathbb{N}\), such that for all \(k>N\), \(d(x_k, x_0)<\epsilon<r\).

Thus, there exist \(x_k\)’s in the \(r\)-neighbourhood of \(x_0\). **Thus, for \(k>N\), \(x_k \in S\), which contradicts our initial assumption that \((x_k) \in S'\).**

Thus, \(x_0 \in S'\).
Since \((x_k)\) is an arbitrary sequence in \(S'\), \(S'\) contains the limit points of all sequences within it.

**Hence \(S'\) is closed.**

\[\blacksquare\]
#### Let \(x,y \in \mathbb{R}\). If \(y-x>1\), then show there exists \(z \in \mathbb{Z}\) such that \(x<z<y\).

**Proof:**

Consider the set \(U=\{u:u<y, u \in \mathbb{Z}\}\).

Since \(U\) is bounded from above by \(y\), it has a least upper bound, call it \(U_\text{sup}\).

We note that \(y-U_\text{sup}<1\). This is because if \(y-U_\text{sup} > 1\), then \(y-(U_\text{sup}+1) > 0\), implying the \(U_\text{sup}\) is not the largest \(x \in U\) which satisfies \(x<y\), which is a contradiction.

Thus, we can write:

\[y-U_\text{sup}<1 \\
\Rightarrow U_\text{sup}-y>-1\]
Adding the above identity to \(y-x>1\), we get:

\[U_\text{sup}-x>0 \\
\Rightarrow U_\text{sup}>x \\
x<U_\text{sup}<y\]
Thus, we have found an \(z \in \mathbb{Z}\) which satisfies \(x<z<y\).

You can prove the same thing by assuming \(V=\{v:v>x, x \in \mathbb{Z}\}\) and taking \(\text{inf } V\), and performing a similar procedure.

\[\blacksquare\]

tags: *Mathematics* - *Proof* - *Analysis* - *Pure Mathematics*