I'm practicing my proof-writing and was hoping you could let me know if this proof looks good. I would like to know if the proof is incorrect, if there are parts that are overly wordy/complicated, or if I'm missing some element of a proof that is helpful to see, if not strictly necessary.
The Prompt:
If $(x_n)$ is increasing, then $lim$ $x_n$ exists (but could be $+∞$). Thus, such a sequence converges if and only if it is bounded above. Show this.
My Proof:
($\to$) Suppose $(x_n)$ is an increasing sequence, and suppose it is convergent. This means that $lim (x_n) = lim$ $sup (x_n) = lim$ $inf (x_n)$ exists and is a real number. $lim$ $inf(x_n) = $$sup_{m}inf_{n\ge{m}}$ $x_n$, and since $(x_n)$ is an increasing sequence, $inf_{n\ge{m}}(x_n) = x_m$ for any $m∈N$. Thus, $lim$ $inf(x_n) = sup (x_n)$, which is only a real number if $(x_n)$ is bounded above, so $(x_n)$ must be bounded above.
($\leftarrow$) Suppose $(x_n)$ is bounded above at $b$, so $sup(x_n) = b$. From before, we know $lim$ $inf(x_n)$ $= sup(x_n)$. Thus,
$lim$ $inf(x_n) = b = inf(b) = inf_{m}(sup_{n\ge{m}}(x_n)) = lim$ $sup(x_n)$
Since $lim$ $inf(x_n) = lim$ $sup(x_n)$ $= b$, $(x_n)$ converges to $b$.