A divergent sequence can have a convergent subsequence.
\"", "name": "tr8", "description": ""}, "x": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "x", "description": ""}, "f1": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"A bounded sequence is convergent.
\"", "name": "f1", "description": ""}, "ch2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(u=1,tr4,if(u=2,tr5,tr6))", "name": "ch2", "description": ""}, "f20": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'It is not possible for an unbounded sequence to have a bounded subsequence.'", "name": "f20", "description": ""}, "f4": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"If {$\\{x_n\\}$} converges, then {$\\{x_{n+i}\\}$} could diverge for some natural number $i$.
\"", "name": "f4", "description": ""}, "f3": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"A convergent sequence is either increasing or decreasing.
\"", "name": "f3", "description": ""}, "tr2": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"If {$\\{x_{n+i}\\}$} diverges for some natural number $i$, then {$\\{x_n\\}$} diverges.
\"", "name": "tr2", "description": ""}, "f9": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'There exists a sequence that is not bounded but which converges.'", "name": "f9", "description": "There exists a sequence that is not bounded but which converges.
"}, "tr11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A finite set is bounded.', 'A set that is not bounded has an infinite number of elements.')", "name": "tr11", "description": ""}, "tr3": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"If a sequence is not bounded, then it does not converge.
\"", "name": "tr3", "description": ""}, "tr6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A sequence with only a finite number of non zero terms converges to 0.'", "name": "tr6", "description": ""}, "ch8": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(x=1,f11,if(x=2,f12,if(x=3,f13,f14)))", "name": "ch8", "description": ""}, "f13": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded set never has a maximum element.', 'A bounded set never has a minimum element.')", "name": "f13", "description": ""}, "f14": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A set with an infinite number of elements cannot be bounded.'", "name": "f14", "description": ""}, "tr1": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"A convergent sequence is bounded.
\"", "name": "tr1", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "g", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "t", "description": ""}, "w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "w", "description": ""}, "f11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A set with a maximum element is necessarily bounded.', 'A set with a minimum element is necessarily bounded.')", "name": "f11", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "v", "description": ""}, "ch1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(t=1,tr1,if(t=2,tr2,tr3))", "name": "ch1", "description": ""}, "tr10": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A set with a maximum element is necessarily bounded above.', 'A set with a minimum element is necessarily bounded below.')", "name": "tr10", "description": ""}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "u", "description": ""}, "tr12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A bounded set has both a least upper bound and a greatest lower bound.'", "name": "tr12", "description": ""}, "ch6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(h=1,f7,if(h=2,f8,if(h=3,f9,f10)))", "name": "ch6", "description": ""}, "f7": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"All convergent sequences of positive terms converge to a value $> 0$.
\"", "name": "f7", "description": ""}, "f12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded set necessarily has a maximum element.', 'A bounded set necessarily has a minimum element.')", "name": "f12", "description": ""}, "tr9": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"There exists a sequence with all terms greater than zero and with limit 0.
\"", "name": "tr9", "description": ""}, "h": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "h", "description": ""}, "f2": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"Every divergent sequence is unbounded.
\"", "name": "f2", "description": ""}, "tr20": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'If a sequence has the subsequence given by the even terms converges to the same limit as the subsequence of odd terms, then the sequence also converges to that limit.'", "name": "tr20", "description": ""}, "f10": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'It is not possible for a sequence to be both increasing and decreasing.'", "name": "f10", "description": ""}, "tr5": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"It is possible for a sequence to be both increasing and decreasing.
\"", "name": "tr5", "description": ""}, "f5": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"There exists a convergent sequence {$\\{x_n\\}$} with $x_n >0$ for all $n \\\\in \\\\mathbb{N}$ and limit $\\\\ell <0$.
\"", "name": "f5", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "f", "description": ""}, "ch7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(w=1,tr10,if(w=2,tr11,tr12))", "name": "ch7", "description": ""}, "f8": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"If {$\\{x_n\\}$} diverges, then {$\\{x_{n+i}\\}$} could converge for some natural number $i$.
\"", "name": "f8", "description": ""}, "tr7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded increasing sequence converges.','A bounded decreasing sequence converges.')", "name": "tr7", "description": ""}, "f6": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"There exists a bounded increasing sequences that does not converge.
\"", "name": "f6", "description": ""}, "ch3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(v=1,tr7,if(v=2,tr8,tr9))", "name": "ch3", "description": ""}, "ch4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(f=1,f1,if(f=2,f2,f3))", "name": "ch4", "description": ""}, "tr4": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"In a convergent sequence, all subsequences converge to the same limit.
\"", "name": "tr4", "description": ""}, "ch5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(g=1,f4,if(g=2,f5,f6))", "name": "ch5", "description": ""}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "f7", "f8", "f9", "t", "tr9", "tr8", "tr1", "u", "tr3", "tr2", "tr5", "tr4", "tr7", "tr6", "tr20", "g", "f", "h", "ch1", "ch2", "ch3", "ch4", "ch5", "ch6", "v", "f10", "f20", "w", "x", "f11", "f12", "f13", "f14", "tr10", "tr11", "tr12", "ch7", "ch8"], "name": "True/false statements about limits of sequences", "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (4 correct out of 8) covering properties of convergent and divergent sequences and boundedness of sets. Selection of questions from a pool.
"}, "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"displayType": "radiogroup", "showCellAnswerState": true, "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": true, "showFeedbackIcon": true, "showCorrectAnswer": true, "minAnswers": 0, "minMarks": 0, "shuffleAnswers": false, "variableReplacements": [], "layout": {"type": "all", "expression": ""}, "choices": ["{Ch1}
", "{Ch2}
", "{Ch3}
", "{Ch4}
", "{Ch5}
", "{Ch6}
", "{Ch7}
", "{Ch8}
"], "extendBaseMarkingAlgorithm": true, "matrix": [[1, -1], [1, -1], [1, -1], [-1, 1], [-1, 1], [-1, 1], ["1", "-1"], ["-1", "1"]], "customMarkingAlgorithm": "", "unitTests": [], "marks": 0, "scripts": {}, "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "answers": [true, false], "warningType": "none"}], "prompt": "\n \n \n[[0]]
\n \n \n \n", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "Answer the following question on sequences and sets. Note that a sequence is said to be unbounded if it is not bounded.
\nNote that every correct answer is worth 1 mark, but every wrong answer loses a mark.
", "tags": ["bounded sequences", "bounded sets", "checked2015", "convergence", "convergent sequences", "decreasing sequence", "divergence", "divergent sequences", "increasing sequence", "limits", "monotone sequence", "monotonic sequence", "sequence", "sequences", "subsequence", "tested1", "unbounded sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "You should be able to work out the correct answers from your notes.
", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}