Let (X, d) be a metric space. Show that every Cauchy sequence {x_n} X is bounded. Solution we need to prove that every cauchy sequence {xn} in metrix space {X,d} is bounded. let assume that xn be a Cauchy sequence of (X, d) hence we can say that by definition of cauchy sequence = 1 so we can say that there exists N such that, d(xm , xn) < 1 for all m, n N hence we can say that xn B(xN , 1) for all n N now r = 1 + max{1, d(x1, xN), d(x2, xN), . . . , d(xN-1, xN)} we can see that, xn B(xN ; r) for all n so we can say that xn is bounded..

1. On a quiz that I'm using to prepare for my test I made a 1.3/2 points on this section so I missed 7
out of the 20 on this section but it doesn't tell me which ones I missed. Please help me figure out
which I got wrong so that I can know for my test. Thank you!!
1) Depolarizing or Hyperpolarizing the Resting Membrane Potential
I put: electrical communication
2) Sum of Exergonic and Endergonic Chemical Rxs
I put: homeostasis
3) Nervous, Epithelium, Muscle, Connective
I put: primary tissues
4) Nervous, Integument, Cardiovascular, Urinary
I put: organ systems
5) Answering Physiological Questions through Experiments
I put: scientific methods
6) Saturated and Unsaturated Fatty Acids
I put: non-polar macronutients
7) Cytoskeleton, ER, Ribosome, Golgi
I put: cell structures
8) Distribution of Osmolarity between the Extracellular and Intracellular Spaces
I put: plasma membrane structures
9) Stable Physiology Through Adaptive and Corrective Process
I put: homeostasis
10) Membrane Channels allow Polar, Small Molecule Movement toward Chemical Equilibrium
I put: plasma membrane structures
11) Biological Catalysts: Name always Ends with -ase
I put: enzymes
12) Anaerobic Catalysis of Glucose
I put: glycolysis
13) Fatty Acid Catabolism to Acetyl CoA, NADH, and FADH2
I put: B-oxidation
14) Catalysis of Acetyl CoA to CO2, NADH, and FADH2 in the Mitochondria
I put: aerobic ATP synthesis
15) Synthesize DNA during Cell Cycle S-Phase
I put: transcription
16) Maintain Vital Total by Adjusting Input, Output, and Production
I put: mass balance

2. 17) Making molecules move UP their Gradient across the Plasma Membrane
I put: active transport
18) Electrolytes, Minerals, Vitamins
I put: enzymes
19) Cholesterol, Glycophospolipids, Integral and Associated Proteins
I put: non-polar macronutrients
20) Peptide Bond Synthesis linking Amino Acids to Form a Peptide
I put: semi-conservatice replication
These are the choices I have to chose from for each of these:
Facilitated Diffusion, B-Oxidation, Homeostasis, Fluid Compartments, Plasma Membrane
Structures, Active Transport, Enzymes, Electrical Communication, Aerobic ATP synthesis, Mass
Balance, Primary Tissues, Non-Polar Macronutrients, Organ Systems, Transcription, Cell
Structures, Micronutrients, Scientific Method, Semi- Conservative Replication, Glycolysis,
Metabolism
Solution
Answers:
1. Electrical Communication
2. Metabolism
3. Primary tissues
4. Organ system
5.Scientific method
6. Non-polar macronutrient
7. Cell structures
8. Fluid compartments
9. Homeostasis
10. Facilitated diffusion
11. Enzymes
12. Glycolysis
13. B-oxidation
14. Aerobic ATP synthesis
15. Semi-conservative replication
16. Mass balance
17. Active transport
18. Micronutrients

3. 19. Plasma membrane structures
20. "Translation" (not present in the options). None of the options match.