calculus-1-homework.ipynb

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   "source": [
    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basic setup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create anaconda environment\n",
    "<br>\n",
    "```bash\n",
    "conda create -n ml python=3.7.4 jupyter\n",
    "```\n",
    "Install fastai library\n",
    "<br>\n",
    "```bash\n",
    "conda install -c pytorch -c fastai fastai\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "!pip install https://github.com/ipython-contrib/jupyter_contrib_nbextensions/tarball/master\n",
    "!pip install jupyter_nbextensions_configurator\n",
    "!jupyter contrib nbextension install --user\n",
    "!jupyter nbextensions_configurator enable --user"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Metric spaces"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that:\n",
    "For every metric space (X, d):\n",
    "- 1.1 For eny $\\mathcal{U} = \\{U | U$ is open in $(X, d)\\}$ holds $\\bigcup_{U \\in \\mathcal{U}} U$ is open in $(X, d)$\n",
    "- 1.2 For eny two $U, V \\subset X$ open in $(X, d)$ holds: $U \\cap V$ is open in $(X, d)$\n",
    "- 1.3 $X$ is open in $(X, d)$\n",
    "- 1.4 $\\emptyset$ is open in $(X, d)$"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "1.1 Let $I$ be indexing set of $\\mathcal{U}$\n",
    "<br>\n",
    "Let $U_i$ be open in $(X, d)$ for $\\forall i \\in I$\n",
    "<br>\n",
    "Let $x \\in \\bigcup_{i \\in I} U_i$\n",
    "<br>\n",
    "Then $\\exists k \\in I : x \\in U_k$\n",
    "<br>\n",
    "Since $U_k$ is open in $(X, d):$\n",
    "<br>\n",
    "$\\exists\\epsilon > 0 : B_{\\epsilon}(x) \\subseteq U_k$ where $B_{\\epsilon}(x)$ is open $\\epsilon$-ball in $(X, d)$\n",
    "<br>\n",
    "$U_k \\subseteq \\bigcup_{i \\in I} U_i$\n",
    "<br>\n",
    "$\\exists\\epsilon > 0 : B_{\\epsilon}\\left ({x} \\right) \\subseteq \\bigcup_{i \\in I} U_i$\n",
    "<br>\n",
    "$\\blacksquare$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.2 Is a special case of:  For eny finity set $(U)_{i=1}^{n}$ of open sets in $(X, d)$, $\\bigcap_{i=1}^{n}U$ is open in $(X, d)$, so only proof of general case will be given below"
   ]
  },
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   "source": [
    "1.3 Let $x \\in X$\n",
    "<br>\n",
    "open ball $B_\\epsilon \\left({x} \\right ) \\subseteq X$ by definition\n",
    "<br>\n",
    "$\\blacksquare$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1.4 BY definition open set is set, with all points belonging to open ball conained within that set.Empty set has no points, so result follows vacuously.\n",
    "<br>\n",
    "$\\blacksquare$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2. Prove that:\n",
    "- For eny finity set $(U)_{i=1}^{n}$ of open sets in $(X, d)$, $\\bigcap_{i=1}^{n}U$ is open in $(X, d)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $x \\in \\bigcup_{i=1}^{n}U_i$\n",
    "<br>\n",
    "Then $x \\in U_i$ for all $i \\in \\left[{1 \\,.\\,.\\, n}\\right]$ from intersection definition\n",
    "<br>\n",
    "So $\\exists \\epsilon_i > 0: B_{\\epsilon_i} \\left({x}\\right) \\subseteq U_i$ because all sets are open in (X, d)\n",
    "<br>\n",
    "where $B_{\\epsilon_i} \\left({x}\\right)$ is  $\\epsilon_i$ open ball of x\n",
    "<br>\n",
    "Let $\\epsilon = \\min_{i \\mathop = 1}^n \\left\\{{\\epsilon_i}\\right\\}$\n",
    "<br>\n",
    "$B_\\epsilon \\subseteq B_{\\epsilon_i} \\subseteq U_i$ for all $i \\in \\left[{1\\,.\\,.\\, n}\\right]$\n",
    "<br>\n",
    "$B_\\epsilon \\left({x}\\right) \\subseteq \\bigcap_{i \\mathop = 1}^n U_i$\n",
    "<br>\n",
    "$\\blacksquare$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3. Prove that for set $U \\subset X$ is open if and only if for each point $x \\in U$ there exists the open neighbourhood $V$ of $x$ such that $V \\subset U$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If open neighbourhood does not exist for some point x, nor does exist open ball centerd in x. If for some $x \\in U$ open ball does not exist than U is not open (from open set definition)\n",
    "<br>\n",
    "If for all $x\\in U$ exists open neighbourhood than open ball also exists for each x. If for all $x\\in U$ exists open ball than U is open (from open set definition)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 4. Prove that, closed ball is closed subset in $(X, d)$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $B(x, r)$ be closed ball in $(X, d)$\n",
    "<br>\n",
    "If $B(x, r)$ is closed set, than $X \\setminus B(x, r)$ should be open, so it is enough ito prove $X \\setminus B(x, r)$ is open\n",
    "<br>\n",
    "Let $y \\in X \\setminus B(x, r)$\n",
    "<br>\n",
    "Then $d(y,x) > r$, so $d(y,x_0) - r > 0$\n",
    "<br>\n",
    "Let $r_1 = d(y,x)-r$\n",
    "<br>\n",
    "We need to prove that $B(y, r_1) \\subseteq X\\setminus B(x,r)$\n",
    "<br>\n",
    "Let $z \\in B(y,r_1)$\n",
    "<br>\n",
    "By triangle inequality: $d(x_0,y) \\leq d(x_0,z) + d(z,y) \\implies d(z,x_0) \\geq d(x_0,y) - d(z,y) > d(x_0,y) - r_1 =r$\n",
    "<br>\n",
    "$z\\in X\\setminus B(x,r)$\n",
    "<br>\n",
    "$\\blacksquare$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5. Prove that\n",
    "For every metric space (X, d):\n",
    "- 5.1 For eny $\\mathcal{F} = \\{F | F$ is closed in $(X, d)\\}$ holds $\\bigcap_{F \\in \\mathcal{F}} F$ is closed in $(X, d)$\n",
    "- 5.2 For eny two $F_1, F_2 \\subset X$ closed in $(X, d)$ holds: $F_1 \\cup F_2$ is closed in $(X, d)$\n",
    "- 5.3 $X$ is closed in $(X, d)$\n",
    "- 5.4 $\\emptyset$ is closed in $(X, d)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5.1\n",
    "<br>\n",
    "$X \\setminus \\bigcap_{i \\mathop \\in I} F_i = \\bigcup_{i \\mathop \\in I} ({X \\setminus F_i})$ De morgans law\n",
    "<br>\n",
    "Each of $X \\setminus F_i$ is open because $F_i$ is closed(from closed set definition)\n",
    "<br>\n",
    "Union of finite number of open sets is open (proved earlier), so $ X \\setminus bigcap_{i \\mathop \\in I} F_i$ is open\n",
    "<br>\n",
    "Complement of  $bigcap_{i \\mathop \\in I} F_i$ is open therefore $bigcap_{i \\mathop \\in I} F_i$ is closed (from closed set definiton)\n",
    "<br>\n",
    "$\\blacksquare$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5.2 Is special case of: For eny finity set $(F)_{i=1}^{n}$ of closed sets in $(X, d)$, $\\bigcup_{i=1}^{n}F$ is closed in $(X, d)$ so only proof of general case will be given below"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5.3 Complement of $X$ relative to $X$ is $\\emptyset$, since $\\emptyset$ is open in $(X, d)$ $X$ is closed in $(X, d)$ (from closed set definition)\n",
    "<br>\n",
    "$\\blacksquare$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5.4 Complement of empty set relative to $X$ is $X$. Since $X$ is open in $(X, d)$ empty set is closed in $(X, d)$ (from closed set definition)\n",
    "<br>\n",
    "$\\blacksquare$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that:\n",
    "- For eny finity set $(F)_{i=1}^{n}$ of closed sets in $(X, d)$, $\\bigcup_{i=1}^{n}F$ is closed in $(X, d)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that, if $F \\subset X$ is closed then $X - F$ is open in $(X, d)$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Metrics in Euclidean spaces"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that:\n",
    "- for every $u, v \\in \\mathbb{R}^{n}$: $d(u, v) \\geq 0$\n",
    "- for every $v \\in \\mathbb{R}^{n}$: $d(v, v) = 0$\n",
    "- for every $u, v \\in \\mathbb{R}^{n}$: $d(u, v) = d(v, u)$ (symmetry)\n",
    "- for every $u, v, w \\in \\mathbb{R}^{n}$: $d(u, w) \\leq d(v, u) + d(v, w)$ (triangle inequality)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove the same properties hold for $d(u, v) = ||u-v||_1$ ($||u-v||_1 = \\sum_{i = 1}^{n}|u_i - v_i|$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sequences and limits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that $x = \\lim_{n\\to\\infty}{x_n}$ in $(X, d)$ if and only if (iff) for every $r \\in \\mathbb{R}$ there exists $n_0 \\in \\mathbb{N}$ such that: $x_i \\in B(x, r)$ for every $i \\gt n_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that if $x = \\lim_{n\\to\\infty}{x_n}$ and $x \\notin \\{-\\infty, \\infty\\}$ then $(x_i)_{i=1}^{\\infty} = (x_1, x_2, \\dots, x_n)$ is a Cauchy sequence\n",
    "<br>\n",
    "For closed set $F \\subset \\mathbb{R}^n$ and convergent sequence $(x_i)_{i=1}^{\\infty} = (x_1, x_2, \\dots, x_n)$ such that there exists $n_0 \\in \\mathbb{N}$ such that $x_i \\in F$ for each $i \\gt n_0$ then: $\\lim_{n\\to\\infty}{x_n} \\in F$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that if $F$ is open from previous example, statement does not hold. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that inherited metric is a metric"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits of functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $f:S \\to Y$ is function between subset $S \\subset X$ of a metric space $(X, d_x)$ and metric space $(Y, d_Y)$\n",
    "- We say that the limit of function $f:(S, d_x) \\to (Y, d_Y)$ between metric spaces in some limit point $c \\in X$ of the subset $S$ is $y \\in Y $if for each open neighborhood of $y \\in V \\subset Y$ there exists the open seighborhood of $c \\in U \\subset X$ such that $f(U \\cap S) \\subset V$\n",
    "<br>\n",
    "This definition is equiualent of definition:\n",
    "<br>\n",
    "- The limit of function $f:(S, d_X) \\to (Y, d_Y)$ between metric spaces in limit point $c \\in X$ of the subset $S$ is $y \\in Y $ if for each open ball $B(x, r) \\subset Y$ there exists the ball $B(c, l) \\subset X$ such that $f(B(c, l) \\cap S) \\subset B(y, r)$\n",
    "<br>\n",
    "or\n",
    "<br>\n",
    "- The limit of function $f:(S, d_X) \\to (Y, d_Y)$ between metric spaces in limit point $c \\in X$ of subset $S$ is $y \\in Y $ if for any $r \\in \\mathbb{R}$ there exists $l \\in \\mathbb{R}$ such that for every $x \\in S$ with $d_X(x, c) < l$ implies that $d_Y(f(x), y) < r$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that this three definitions are equiualent for eny function between eny two metric spaces"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Continuous functions "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that function is continuous in $c$ if for eny sequence $(x_n)_{n=1}^{\\infty} \\subset X$ such that $\\lim_{n \\to \\infty}x_n = c$ we have $\\lim_{n \\to \\infty}f(x_n) = f(c)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that function is continuous if for every open set $V \\subset Y$ the $f^{-1}(V)$ is open in $X$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that function is continuous if for every closed set $F \\subset Y$ the $f^{-1}(F)$ is open in $X$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Prove that any composition of continous functions is continous"
   ]
  },
  {
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